It appears that the comments have addressed the question. However, since the comments didn't point to any existing JSE threads, and since I cannot find one that directly talks about this construction, I'll add in a proper explanatory answer post for future reference.
This gets a bit long, out of necessity. :)
Stem-form + する - How does this work?
The sample text 「お話ししましょう」 clarifies that this should be a bit more specifically:
お + Stem-form + する - How does this work?
This is the basic way of constructing the humble verb form.
Most of us probably encounter the phrase おねがいします very early on in our Japanese language studies. This is constructed the same way.
お + ねがい (the stem of verb ねがう, "to wish for something") + します
For the question's sample text:
お + はなし (the stem of verb はなす, "to talk, to speak") + します
While we're at it, and since it's very similar in construction, let's also look at one way of creating an respectful verb form.
At a simpler level, we can use a construction very similar to the humble form above -- all we need to do is replace the します with に + なります.
お + はなし (stem form of verb はなす, "to talk, to speak") + に + なります
... but why would we even use either of these? For that, we have to dive into something called keigo.
Keigo or "honorific language"
Both "humble" and "respectful" forms are part of what is called 敬語【けいご】 (keigo, "honorific language").
Keigo is simply part of how the Japanese language is organized.
For English, one of the key organizing factors is who is doing the action. Verbs inflect or conjugate (change form) based in part on the identity of the actor: is it "I", "he / she / it", or "you / we / them"?
I am, he / she / it is, you / we / them are
For Japanese, one of the key organizing factors is who is talking to whom, and about whom. Verbs infect or conjugate based in part on the identity of the speaker and the actor in relation to the listener.
する (friends, casual), します (colleagues or strangers, polite), いたします (strangers, formal, about themselves or their group), なさいます (strangers, formal, about the listener or their group)
If you're at all familiar with other European languages, most of these still have something roughly analogous to keigo, albeit simpler -- the distinction between the two forms of "you". French has its "tu" and "vous", Spanish its "tú" and "usted", German its "du" and "Sie", and Czech its "ty" and "vy".
English used to have this, back in Elizabethan times: "thou" was both the singular and the intimate / casual, while "ye" was the plural and the formal / polite.
The basics of verb forms and "politeness"
When English speakers are taught beginning Japanese, pretty much everyone starts with the so-called "‑masu form" of verbs, things like します ("to do") and はなします ("to talk, to speak") and です (the basic "to be" verb: "is / am / are"). This is often called 丁寧語【ていねいご】 ("polite language"), and it's probably the most "neutral" verb form, appropriate in most social circumstances.
Later on, teachers will introduce the so-called "plain form" of verbs, things like する and はなす and だ. This is the basic form of verbs and is used for dictionary listings. This is also a more "intimate" or "close-in" verb form, appropriate for use among friends or closer acquaintances, people who are part of your closer social group.
Broadly speaking, keigo involves the more "formal" or "distanced" verb forms, appropriate when talking to people more clearly outside of your own social group.
The basics of keigo
Lots of English-language materials I've seen over the years talk about "politeness" and relative social superiority. However, "up" and "down" isn't really the right way to look at it -- the social deixis of Japanese grammar is much more about "inside" and "outside", with respect to whether someone else is part of the same group, and how intimate / close they are. When talking to someone who is definitely outside of your group, such as when talking to a customer, someone from another school or company, or even in a formal setting, you would possibly use keigo. This is essentially the next level that is "more polite" or "more formal" than the "‑masu form".
In-group and out-group
When using keigo to talk to someone from outside your group, you use different verb forms to talk about an "inside person", such as yourself or someone else within your group, as compared to talking about an "outside person", such as the listener or someone else from their group or some other group.
In Japanese, the forms used to talk about an outside person are called 尊敬語【そんけいご】 ("respectful language"), and the forms used to talk about an inside person are called 謙譲語【けんじょうご】 ("humble language").
Complicated? You betcha!
Keigo can get super complicated, with lots of "suppletive" verb forms. (Suppletion is where a totally different word or root shows up and supplements or adds on a form or grammatical function that was missing, or replaces an existing form. See the linked Wikipedia article for more.) One common case of honorific suppletion is turning the verb いきます ("to go") into completely unrelated verb いらっしゃいます ("to go") for respectful use, or another completely unrelated verb まいります ("to go") for humble use.
→ If you're worried about using keigo correctly, you're not alone! It's complicated and easy to get wrong. There are even books about it written in Japanese for Japanese speakers to learn how to use keigo more appropriately. The お +
[VERB STEM] + します or なります constructions described above might be the easiest way to get into keigo.
As with anything, study and practice will help. Good luck! 😄