"Budgie Smugglers" is an Australian colloquial term given to a particular style of bathing suit. Also known as speedos or banana hammocks, these tight, briefs style, lycra bathing suits are worn by men. The combination of being worn by men and being small & tight fitting causes a noticable bulge at the front that looks like, as it were, the man is smuggling budgie/s in his swimsuit.
There is a side inference that the guy wearing the budgie smugglers is a bit ridiculous. It is laughable to imagine someone could attempt to smuggle small birds by stuffing them into a tiny bathing suit they are wearing, where the bulge caused by the birds makes it obvious that one has done so. So too, one tends to laugh a little at someone who has chosen swimwear that emphasises the bulge so obviously by barely covering it. It is generally more acceptable for men in social swimming situations to wear board shorts instead.