One of the many uses of the comma is to add an aside to a sentence. The aside is a secondary part to the sentence, one that may add additional, even important, information but is not the main part of the sentence itself.
Here's the sentence when we remove the asides:
All my childhood I longed desperately for a tricycle, and later the wish was transferred to a bicycle.
Simple enough. The person wanted a tricycle as a child, and later on wanted a tricycle. Now let's add the first aside back in:
All my childhood I longed desperately for a tricycle, which my parents could never afford, and later the wish was transferred to a bicycle.
This is a good piece of additional context, because it explains why the person didn't have a tricycle as a child: because his/her parents couldn't afford one. However, it isn't the primary part of the sentence, as you can see in the first quote. You can have a perfectly fine sentence without that aside, but by being contained within the commas it can be neatly tucked in with it.
The second aside works for the same reason: it adds another piece of potentially important context (that his/her parents couldn't afford a bicycle either), but it's still secondary to the main point.
(I'll admit that the quote referenced is pretty convoluted in how it's worded, so I don't blame you for getting a bit lost here. I'd agree with Peter with adding a comma after the "too", because it looks a bit jumbled without it.)