Some skills do affect combat. As others have mentioned, Acrobatics is immensely useful in combat for getting past tanks who block choke points and for avoiding attacks of opportunity.
Some skills won't affect combat. They don't need to. They affect other things.
In the olden days (AD&D and before), monster manual entries were generally held to be just something to come across in a dungeon room, kill, loot, and move on to the next one. This actually caused problems if you wanted to interact with them in literally any other way than this pattern, because the rules simply didn't support it. The issue wasn't so severe as it might sound, because DMs were expected to wing it, but it did unnecessarily make the DM's job that much harder than it already was, and the asymmetry of interacting with most demihumans by one set of mechanics, and a completely different set of mechanics for interacting with "monsters", was rather jarring for many, myself included.
The introduction of 3.0 changed this by giving monster manual entries all the same types of stats as player characters: not just hit points, saves, AC, attack bonus and attack damage, but also actual base scores in all 6 abilities (Str, Dex, etc), skills (not only combat-relevant ones, but also utility and social), spellcasting progressions that included the possibiity for utility spells, and so on. This makes it possible for lying to a monster, sneaking up on a monster, being snuck upon by a monster, breaking a monster's concentration, and determining what a monster gets how much better at when you buff his Int with a spell, to all be covered by the same rules that cover those things happening with PCs instead of monsters. As a 3.x derivative, Pathfinder inherited this change.