Bioavailability is basically a measurement indicating how much of a drug is absorbed into the blood stream... Most drugs when used orally (eaten) undergo metabolism (breakdown) when they pass through the intestines, and when they go straight to the liver... If a drug is injected, it doesn't suffer any metabolism before it reaches the blood (because it is injected straight into the blood), so when a drug is IVed you can say it has a bioavailability of 1. Conversly, if when you ate a drug, it was completely destroyed in the liver, it would have a bioavailability of 0.
There are all kinds of ways to calculate a drugs bioavailability, but thats not really important. How using a drug sublingually increases bioavailability is because, of course, the drug passes through the membranes of the mouth, straight into the blood, without being metabolised in the liver first.