Meaning of quedar
Quedar has several meanings.
Here, it means
3 (to set a date or time)
a. to meet
¿Dónde quedamos? Where shall we meet?
b. to agree to meet
Quedamos en la puerta principal. We agreed to meet at the front door.
c. to arrange to meet
He quedado con mi abogado a las 10 am porque quiero pedir consejo sobre el tema. I've arranged to meet my lawyer at 10 am because I want to ask him for advice on the matter.
How does the usage of the present perfect tense compare to English? Wikipedia provides this guidance (notice especially the last sentence):
The present perfect is a grammatical combination of the present tense and perfect aspect that is used to express a past event that has present consequences. The term is used particularly in the context of English grammar to refer to forms like "I have left."
In English, completed actions in many contexts are referred to using the simple past verb form rather than the present perfect.
There are three instances of present perfect tenses in your example. Let's see how they might translate to English:
Te he llamado porque he quedado con unos amigos para tomar un café esta tarde y he pensado que podías venir con nosotros, ¿Te apetece?
I'm calling because I've arranged to have coffee with some friends this afternoon and I was thinking that you could come with us. Does that appeal to you?
We know that the present tense in Spanish can be used to talk about the near future (e.g. La semana que viene regreso a los Estados. Next week I'm returning to the States; Mañana me pagan Tomorrow I'll get paid).
The Spanish simple present tense can be used to talk about habitual actions, routines, things happening now or in the near future, universal truths, facts, hypotheticals, lapses of time, and for ordering in restaurants and stores. (spanishdict.com)
So actually, when Wikipedia wrote
used to express a past event that has present consequences
they could actually have said
used to express a past event that has present or near future consequences.
That's what's happening with the projected coffee date for this afternoon in your sentence.
I speak Latin American Spanish (Mexico). I've noticed that at least some people from Spain use the present perfect more often than is common in Mexico. (I don't know if there are regional differences within Spain.)
For example, I've noticed that friends from Spain would be more likely to say
La carta ha llegado esta semana The letter arrived this week
where I'd be more likely to say
La carta llegó esta semana The letter arrived yesterday
(I'm reconstructing the Spain version from memory so if it's not quite right I hope someone from Spain will correct it.)
Your example would be perfectly understandable in Mexico, but I'd be more likely to express it something like this:
Te llamaba porque que quedé con unos amigos de mi grupo de teatro para tomar un café y pensaba que quizás te gustaría conocerlos. ¿Se te antoja?
I was calling because I'll be meeting up with some friends from my theater group at a café and I was thinking you might like to meet them. You want to? [Does that appeal to you? / Do you feel like it?]
(One slightly tricky thing here is that arranging to have coffee is code language in Mexico for setting up a first date with a romantic interest. So I made a few other little changes to make a more neutral invitation -- as opposed to letting it sound like the speaker is hitting on the other person.)