The thing is, saying that infinity + any number equals infinity is a bit imprecice.
There are two widely used concepts of infinity; they refer to cardinality and ordinality.
In finite numbers, Cardinality is the concept of "how many" of something there are -- 1 sheep, 2 sheep, 3 sheep. Ordinality is "what order" they come in -- 1st sheep, 2nd sheep, 3rd sheep.
With finite numbers they are highly tied to each other. You can just "count labels" in a sense.
With infinite sets the two concepts diverge.
We'll start with the natural numbers -- the set of all counting numbers. 0, 1, 2, 3 etc. That'll be our "first infinity".
If you take the first infinity, and add another element to it, you get the same cardinality. This is what people talk about when they say "infinity+1 equals infinity". More than that, if you take the first infinity, and double it, you get ... the same cardinality. You can even add an infinite number of infinities to it -- take "first infinity times first infinity" or "first infinity squared" -- and you get the same cardinality.
It isn't until you reach (assuming continuum hypothesis) 2^"first infinity" that you reach a new cardinal. This can either be described as the "set of all the first infinity" or "the set of functions that go from the first infinity to yes/no" (it shouldn't be hard to see they describe the same thing). This is a bigger cardinal than the first infinity, and is also the same cardinality as the real numbers.
The cardinality game continues from there.
So that is one branch.
The other is ordinals. In ordinals, we talk about ordering things. For any two things, you can say which is in front of the other in the order. And for any collection of things ordered, we can find the "least" element (the one "behind" all the others), including the entire collection of ordered elements (we normally call this element 0).
The "first infinity" in ordinals is ordering everything by the natural numbers. Everyone gets a tag that says "1st" or "1 million and 7th" or whatever.
Now, in ordinals, we can they have someone with the label "1st in 2nd lineup", and we can state that the 2nd lineup goes after every value in the first lineup. This is "infinity plus 1" in ordinals, and it is a distinctly different way of ordering people.
What more you can have 2 infinite lineups (where the 2nd goes after the first), or 3, or an infinite number of infinite lineups (where each lineup goes after the one before). These are all distinct ordinals -- they describe fundamentally different ways of ordering things.
And the game continues from there.
Now that I have disabused you of the notion that infinity+1 always equals infinity, how do we talk about it with a 5 year old?
You could talk about that split. Say "up to infinity the idea of ordering and counting is the same. At infinity they are different."
Then talk about infinite ordering and lineups.
The ordinal $\omega$ is a lineup that goes on forever. There is someone in front.
The ordinal $\omega + 1$ is two lineups. One that goes on forever, and one with a single person in it. That single person goes after the first lineup. It will get very boring for them.
The ordinal $\omega +2$ has 2 people in the second lineup.
The ordinal $\omega + \omega = 2 \omega$ has two infinitely long lineups. The second goes after the first.
The ordinal $k \omega$ has $k$ lineups, each infinitely long.
The ordinal $\omega \omega = \omega^2$ has an infinite number of lineups, each infinitely long.
The ordinal $\omega^2 + 1$ has an infinite number of lineups, each infinitely long, plus one person who gets to go after everyone else is done.
The ordinal $\omega^2 + \omega$ has an infinite number of lineups, each infinitely long, plus another lineup that goes after the previous infinite set of lineups are all done.
The ordinal $2\omega^2$ has a two collections, each with an infinite number of lineups, each infinitely long, with one going after the other.
The ordinal $\omega \omega^2 = \omega^3$ has an infinite number of collections, each with an infinite number of lineups, each infinitely long.
The ordinal $\omega^\omega$ has an infinitely long order of layers. In each layer, there is an infinite number of the next layer, all ordered.
You probably will break down before getting this far.
Also talk about cardinality. Here the other answers cover things really well -- things like the Hilbert Hotel and the Cat in Numberland are great resources.
A fun part of this is that every Cardinality has a whole bunch of Ordinalities associated with it. You can look at the stories, like Hilbert Hotel, and talk about how the Ordinality changed even when the Cardinality didn't.
And you can talk about how this doesn't work for "normal" numbers. You cannot change the fundamental ordinality without changing the cardinality.
Having two lines, one of 2 people, followed by a line of 3 people, is the ordinal 2+3, which is fundamentally the same as the ordinal 5. You can "just paste" the 3 people onto the end of the first line.
With infinite ordinalities, you cannot reach the end of the first line to paste the second line on. It is infinitely far away.