As Timmy Jim stated, and as was mentioned in the question itself: capturing a specific type of Pokémon (not species) will increase your odds of capturing other Pokémon of that type. Each badge will increase your odds by a factor of 1, though it is not yet determined what this means.
- For capturing 10 Pokémon of a given type (bronze badge): +1 bonus catch rate
- For capturing 50 Pokémon of a given type (silver badge): +2 bonus catch rate
- For capturing 200 Pokémon of a given type (gold badge): +3 bonus catch rate
These bonuses will be averaged in the case of a dual-type Pokémon. For instance, if you attempt to catch a Gloom (Grass/Poison) and you have a gold Punk Girl badge (thanks to all those Weedle), but only a silver Gardener badge, you will have a bonus catch rate of +2.5, as the gold badge gives you +3 and the silver gives you +2.
The two prevailing theories as to what these numbers mean at this time are:
The +1/2/3 represent an extra 10/20/30% chance to catch a Pokémon, which would be an added multiplier after the original catch rate calculation. This theory proposes that after catch mechanic calculations have been done, the Final Catch Rate (see here for an explanation of the catch mechanics) is multiplied by 1.30 (assuming a full +3 catch rate bonus).
The +1/2/3 represent an added 1/2/3% increase to the base catch rate of a Pokémon. Under this theory, for hard-to-catch Pokémon such as Dragonite, a gold badge would nearly double its base catch rate (from 4% to 7%).
Though the update is very recent, here is some anecdotal evidence to show how much the bonus catch rate works in practice. Bear in mind, the overall catch rates may have been increased with this update as well, which may slightly skew the data, but the increase in catch rate is very clear.