This question can be answered at two levels. I'm going to take the easy one. I really hope that someone more advanced than me in the lore of algebraic topology can pick up the hard one.

The easy answer is that if the homotopy groups of spheres weren't so complicated then we wouldn't be talking about the homotopy groups of spheres so much.

Let me expand on that by an analogy. A penknife is a useful tool. One can do a lot with a penknife, but there's a lot of things that it's not that good at: getting corks out of bottles, descaling fish, sawing small bits of wood, getting annoying bits of food out from between your teeth ... I mean, I know that you can do a lot of those things with a penknife if it's all you've got, but it's not the best way to achieve those ends. Now a Swiss army knife is much better at doing all those. The latest probably also have inbuild GPS! But Swiss army knives are quite complicated gadgets. So when you say, "Why are Swiss army knives so complicated?" then the easy answer is that if they weren't, we wouldn't be using them so much and we would have found something else that was complicated to use instead.

In slightly less prosaic language, the fact that the homotopy groups of spheres are so complicated is what makes algebraic topology actually useful. We want to build complicated objects out of simple ones. What could be simpler than spheres? But to get something complicated, there has to be a source of complexity (I'm speaking very informally here) otherwise there would be no real hope of algebraic topology ever helping with other things. I mean that we know that general stuff in mathematics is quite complicated, so we're going to need some complicated tools to study it. If the homotopy groups of spheres were simple, then algebraic topology wouldn't be half so useful as it is; and if that were the case then there wouldn't be so many algebraic topologists around and your friend probably wouldn't have heard of the homotopy groups of spheres.

Let me finish with an attempt to clarify what I think is the hard part of this question to answer. That is, "Why spheres?". We accept as given, as I've argued above, that we need a complicated theory to study complicated objects; but the methods of algebraic topology are to probe the complicated objects by simple ones and so, hopefully, for any specific question to get rid of all unnecessary complexity and be able to see clearly the structure required for that specific question (I think that the proof of the Kevaire invariant problem is an example of what I mean here). So we need a good source of "simple objects" to probe with. Now these "simple objects" are those that look simple when we look at them with the tools of algebraic topology. So spheres are simple because they have very simple cohomology.

But we can probe something in two ways: we can either throw mud at it and see what sticks (that's homotopy), or we can take pictures of it and see what it looks like from different angles and with different lighting conditions (that's cohomology). As I've argued, the theory needs to have some complexity somewhere, so it's to be expected that the objects that are simple with respect to one method will look complicated when viewed at from the other. So spheres have complicated homotopy because they have simple cohomology. In contrast, the Eilenberg-Mac Lane spaces have complicated cohomology because they have simple homotopy.

But still, "Why spheres?". I mean, no-one ever asks, "Why do the Eilenberg-Mac Lane spaces have complicated cohomology?". I guess that's because no-one outside algebraic topology ever meets Eilenberg-Mac Lane spaces and so they aren't common objects across all of mathematics. So of course they have complicated cohomology because they are some weird tool that algebraic topologists have constructed and who knows what secret rites were used to do it?

So maybe I do have an answer to my "hard part" of this question: it's historical. In the early days of algebraic topology, the pioneering homotopy theorists got the idea of studying a space by throwing mud at it and seeing what stuck. As this was a new thing to try, they looked for the simplest thing that they could find: spheres. Then they found that they had a useful theory that had enough complexity to study spaces, and this was evidenced by the complexity of the homotopy groups of spheres. Had the homotopy groups of spheres been simple, algebraic topology wouldn't have gotten off the groups and, as I said, your friend would probably never have heard of it or them.

So, in summary, my answer is: something powerful enough to study a space by being thrown at it is going to have some complexity somewhere; spheres were the first thing that people tried, and they proved to be sufficient. (One could continue this by asking: why were spheres enough? But the answer is the same: if they weren't, we would have gone further. Spheres aren't enough to study everything, but they are enough to study most things that people are interested in.)