Phrase used to describe a language partner that simply wants to use the time allocated for language exchange as a private English lesson. This does not mean the person is a "bad person" -- in fact most English leeches are really nice people, and that's part of the problem. In the vast majority of cases language partners will know more English than you know of their mother tongue, and so switching back to English is simply easier for both of you, especially if you like this nice person and want to communicate with them!
Particularly prevalent in Asian countries, as the education systems of many Asian nations emphasize grammar and writing over spoken language learning. Thus, a person allegedly at the same level of proficiency in English as the English speaker's level in the second language in question, will actually have a far greater vocabulary, but simply very little practice in speaking English, which easily unbalances what can be a genuine attempt at language partnering.
It can sometimes be very difficult to get rid of an English leech, due to cultural differences about confrontation and also because, for the English leech, the language exchange is working out fine! But, for optimum language learning, you need to split the time between the two tongues strictly 50/50. Be firm.