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.
Though this doesn't explain exactly how this ambiguous +1/2/3 modifier works, some people have reported that the badge shines occasionally with the "Gotcha!" notification once a Pokémon is captured. (See here and here).
I noticed that sometimes, when you catch a Pokémon which catch rate is modified because of the badges you have, the badges show up again and shine.
I believe this happens because the badges actually had an effect on catching that specific Pokémon - something like "look, you only caught this one because of the badges you have."
If this is an indicator that the badges have helped you capture the Pokémon (which is what many people think), it, at the very least, shows that the badges are doing something to affect your catch rate.
Yesterday, an article was posted by the Pokemon Go GamePress team that had a reasonable estimate on how the medal catch rate bonus works. The working hypothesis is that the +1, +2, and +3 bonuses act as multipliers on the catch chance for an encounter, meaning that there is an additional multiplier on the probability to catch of 1.10 for a bronze medal, 1.20 for a silver medal, or 1.30 for a gold medal.
We also have statistical evidence suggesting that the bonus provided by gold medals is 1.3; applied in a similar matter as other multipliers in the GUCT formula.
The GUCT (Grand Unified Catch Theory) can be seen below:
It is worth noting that this is based on 1147 data points, but 1053 of them are for the gold medal bonus. The results show that the estimated value for the +3 bonus is
1.339 +/- 0.031, but based on how the catch mechanics work, it seems to be a safe bet that the real value is strictly 1.30.
Bear in mind the following disclaimer:
This is clearly just a reasonable theory, not a statistical conclusion, but we consider it likely enough that we'll go forth with those multipliers as we continue our research in this area.