"Bots" is short for robots, and in this context is a computer program that works as an agent for another system or simulates a human being/human action.
Example, I can write a bot that could search for frog pictures on the internet, log into my tumblr, and post one of those pictures for me. Now that I have my fancy bot I can have posts of frog pictures always going up on my tumblr feed without any interaction on my part.
A porn bot is an extension of malicious bots, these are robots that were designed to perform an action that will ultimately generate money for the person who wrote them. Usually illegal activity and to the detriment of innocent victims the bots interact with.
The porn bots I have seen on tumblr specifically attempt to get me to click on a link that probably has a payload that would install some malware on my computer. What can someone do with a hacked computer? Well a whole bunch of stuff.
(I got this image from Krebs on Security, https://krebsonsecurity.com/2012/10/the-scrap-value-of-a-hacked-pc-revisited/, who is a journalist who likes to report on cyber security news)
What a hacker can do with your compromised PC is quite a bit so I am not going to go much further then that in this post, but just know all those items in the above image can be leveraged to steal money.
WHY DOES TUMBLR HAVE SO MANY PORN BOTS?
Honestly the recent influx of these on Tumblr probably relates to the situation happening at Twitter. Tumblr's reputation is on the rise for the first time in awhile, it is having very positive user growth and online attention. This in turn attracts the attention of thieves and gangs looking to make money off the crowd. In much the same way pick-pockets work crowded train stops, scammers need to operate where other humans are congregating.
This is a VERY brief run down on Tumblr's porn bots, but this is a huge topic that people dedicate their whole career to understanding.
So hopefully this answered your general question, let me know if you have another question or want me to get more specific about something.