My wife told me to take the spider out instead of killing it...

My wife told me to take the spider out instead of killing it
Went out. Had a few drinks. Nice guy. He's a web designer

This is a joke about a misunderstanding, it has talking animals in it (sort of), and it has a pun. There are also two punch lines in this joke.

First: the misunderstanding. The wife meant "take the spider outside and release it there", and the husband heard "take the spider out on a date". That misunderstanding occurring is pretty unlikely, so it makes the joke funny.

Second: the spider is a web designer - this is a pun. Spiders make literal spider webs, and a sentient spider could possibly be said to design them, however a web designer is usually someone that designs websites, so there are two possible meanings of the term

Additionally, it is most likely true that if he took the spider out and found it to be a nice guy that the spider talks, because otherwise how would he have determined that the spider is nice and a web designer?