So the sentence you quoted is not very well written. Eyes do not well up. Tears do. It should be "Tears began to well up in his eyes." Otherwise, an emotion may also well up, such as resentment or bitterness.

It means that somebody is crying.

If you look up a.dictionary, it will tell you this: well (verb) [often "well up"] (Of a liquid) rise up to the surface and spill or be about to spill: