A couple of questions?
You don’t get a bigger bag, but you can get more of them to increase storage capacity.
http://wiki.guildwars2.com/wiki/Bag
If you have salvageable trophies, salvage them for materials. If they are gray ‘junk’ trophies, then just sell them to the merchant.
http://wiki.guildwars2.com/wiki/Trophy
That’s great. Thank you
Also worth knowing that some trophies are used to complete ingame heart events, like the logging tools used in Queensdale for the Skrit heart. You can still collect these after completing and use them latter for a quick completion to score the extra XP points/unlock the related vendors on new characters.
Trophies like these stack, so you can have up to 250 in one inventory or bank slot, more if you upgrade your bank through the gemstore. This is why you will find them on the Trading Post for sale – a little timesaver/quality of life thing.
Like Mik said, if they are not used for hearts just sell them to one of the regular vendors/traders in the Cities or Outposts/Camps. Trophies to do this with can’t generally be salvaged, which is how you tell them apart from ones like Leather Straps that are salvage materials you can use in crafting or sell on the TP for coin towards something else you want to buy. Worth checking the TP buy/sell prices regularly as prices go up/down.
Finally, some are just cool to collect as memento’s of your adventures – but only if you have the room