This is a phrase that has arisen a number of times in my various ATC nodes, but with a great deal of effort I have resisted the urge to digress too much from the subjects at hand and discuss it in-depth. It is relatively unclear to me now why this is, since the "own nav" idea is actually fairly straightforward. It is also one of the favourite phrases of a typical radar controller. Some others follow, with translations.

American 09 - basic service.
(I don't have time to provide you with any significant type of ATC service, or I can't be bothered)
American 09 - remain outside controlled airspace, basic service available with so-and-so on <frequency>.
(I don't want to talk to you: sod off)
American 09 - radar service terminated, squawk 7000, freecall so-and-so on <frequency>
(I don't care about you any more: sod off)
American 09 - contact so-and-so on <frequency>
(I'm done with you now: sod off)

Looking at this list, there's actually a lot to be said for doublespeak.

Anyway, "resume own navigation" would definitely be in the above list somewhere; what rank, exactly, depends on the type of controller you're asking. Tower controllers - who work at airfields - for example, would have no use for it whatsoever, because they generally don't use radar. Approach controllers - who deal with aircraft flying near airfields - would probably fall in the middle; they would not use the phrase as often as the remaining category, but would always appreciate it when they do. Area controllers, who deal with relatively high-flying, en-route traffic, would probably put the phrase at or near the top.

To explain the utility of the phrase it is necessary to explain a little about the rules of flight. There are two very broad classifications of airspace: controlled and uncontrolled. Cleverly and respectively titled 'controlled' and 'uncontrolled'. Controlled airspace is subdivided into six categories, designated letters from A to E. Uncontrolled airspace is all the rest, and is variously designated F or G. The degree of control that ATC exercises over a flight diminishes with ascending class of airspace. Class A airspace is a totalitarian state (and the preferred class of most public transport flights) while class G is, like, Sealand or something.

You might also be aware of the two broad categories of rules that flights can operate under - these are the visual flight rules (VFR) and the instrument flight rules (IFR). It's not necessary to go into horrendous detail and the respective nodes will give you more information; for now it's sufficient to say that VFR flights fly with visual references and operate under a "see and be seen" principle. The weather conditions have to be sufficient to allow this kind of flight to take place - you must be able to see a minimum distance, be able to keep away from clouds and be able to see the ground at all times. This also means you can't fly VFR at night.

Needless to say these conditions have to be formalised somewhere in order for the rules to mean anything - we call them "visual meteorological conditions" (VMC). The conditions under which VMC is said to exist do vary slightly by class of airspace, but in all cases if these conditions are not met then we say that "instrument meteorological conditions" (IMC) apply. Now we get to IFR: IFR allows you to fly using only your aircraft's on-board instruments, rather than looking out the window to make sure you're not, for example, upside-down. You need much extra training to get an IFR, or "instrument" rating added to your pilot's licence, but since it permits you to fly in IMC conditions - that is, poor visibility and at night - it adds substantially to your usefulness in the role.

Now we finally get to ATC's role in all this. You might be aware that controllers can tell aircraft which direction to fly in - this is called vectoring, or locking aircraft on a heading. Controllers can give aircraft a specific magnetic bearing to fly, or tell them to turn left or right by a specific amount, with the caveat that it is only permitted in certain classes of airspace, and can only be done to IFR flights. Why not VFR? Because controllers usually can't see weather on their radar screen, and if they were to vector a VFR flight they might inadvertently point it into a cloud, which if the pilot is not instrument-rated, could be very bad. Not to mention illegal.

So, if an aircraft is being vectored, it is flying in the direction ATC has instructed it to. It must fly in this direction, for all intents and purposes, forever. The only condition under which a pilot may divert from their ATC clearance (for this purpose an aircraft's "clearance" is the direction and level it has been instructed by ATC to fly at) is when they feel that it would endanger their flight to follow it. Despite ATC being the overlords of controlled airspace, ultimate responsibility for the safety of a flight always rests with the pilot, and as such they are quite within their rights to tell ATC to get bent if they feel that their instructions are unsafe.

Why do we vector? Either to keep an aircraft safely apart from others, or because we want to exercise fine control over where they are flying. The latter applies most frequently when an aircraft is coming in to land - Approach Control will vector them around so that they are in a position from which they can easily approach the runway and land safely. The former applies when aircraft need to exchange levels: for example, an aircraft wanting to climb through the level of an aircraft above it. As you may be aware aircraft must be a minimum distance apart to be considered 'separated'; this varies slightly by area but is most commonly this:

Aircraft that are separated horizontally by less than five miles must be separated vertically by at least 1,000ft.

So when two aircraft are less than 1,000ft apart vertically, they must be at least five miles apart horizontally. This is frequently achieved using vectors - the two aircraft are instructed to fly in such directions that they are at least five miles apart, then the aircraft underneath can climb 'through' the aircraft above it.

Now, an aircraft may well be following ATC instructions to fly in a particular direction, but the pilot will of course have their own ideas. They will be wanting to fly to a specific destination, and most likely they will have a route to get there nicely planned out before they even leave the ground. When an aircraft is following its preplanned route we say it is on its "own navigation": ATC is not specifically telling it which direction to fly, but is aware of its published routing.

This is all very well, but you can't count on it to provide separation. If two aircraft are on their own navigation, on separate parallel routes that just happen to be five miles apart, they are not separated. ATC work on the assumption that aircraft on their own navigation could divert from their route at any time - to avoid a huge thunderstorm in their path, for example. But if aircraft are locked on headings, those headings are considered "guaranteed".

So we now know how headings are used to separate aircraft, and arrive at the reason that "resume own navigation" is such a beloved phrase.

Vectoring is a pain in the ass. It effectively doubles the workload for an aircraft. If an aircraft is following its own routing, then you don't have to worry about where it is going because it "knows". You only have to make sure it is at an altitude that ensures it doesn't ram something going the other way. But an aircraft on a heading, as I said before, will essentially fly that heading forever. Controlled airspace has specific boundaries, and an aircraft locked on a heading will eventually blunder outside those boundaries if it is left unattended, and subsequently be afforded much less protection from others. A controller has to make sure aircraft on headings are going the right way and adjust them if not, which is an extra check for them to make every time they look at an aircraft.

So when headings cease to be required - that is, aircraft are safely separated by altitude - a controller will almost certainly cease using them and instruct the respective aircraft to "resume own navigation". A good controller is a lazy controller, as an instructor once said.