Mage Slayer Feat states that you have an advantage. So you do. There are no hidden rules.
However strength check to break free from Web is a strength check. It's not called a saving throw, so it isn't. Only dexterity saving throw is a saving throw in this spell — because only it is named a saving throw in the spell description. Again, there are no secret rules.
If you need an in-world explanation, extrapolate from the first sentence of the feat:
You have practiced techniques in melee combat against spellcasters
Maybe you can observe spellcasters to gain intuitive understanding of weaknesses of their spells? Maybe it's the hatred that fuels you? Maybe ancient rituals during training made spells stick to you less? Whatever story you want to tell, you can. As long as the DM is okay with it, of course.
For the dillema, is the 5ft at the moment of casting, or at the moment the save is made, I believe that "spells cast by creatures within 5 feet of you." means it should be measured at the time of casting the spell. This feat isn't very powerful, however. Guides rate it B or C tier (example). Thus, personally I would rule to allow both interpretations and grant advantage if either is true. I don't see it making this feat overpowered. Most of the saves are made at the time of casting anyway, and making sub-optimal feat a bit more useful in edge cases won't ruin your game.