The Downspout
The downspout is the first thing to freeze and the thing that splits, because water sitting in the bottom of it has nowhere to go.
Short answer
Water standing in the bottom of a downspout or in the extension freezes first, and the ice then backs up the pipe. The expansion splits the seam, usually at an elbow. Keeping the discharge clear and sloped away, and getting the trough cleared in the fall, prevents most of it.
Why they freeze from the bottom
Water sitting in the bottom elbow or in a flat extension cannot drain, so it freezes first.
That plug then stops anything above it draining, so the next meltwater sits on top of it and freezes too.
The column builds upward until the whole downspout is solid.
Expansion splits the seam, and on a downspout the seam is the weak line.
A buried extension that is blocked or has frozen underground does the same thing and is invisible.
And a discharge onto a paved surface that is itself iced over has nowhere to go.
The split is at the elbow
Look at the elbows after a freeze, particularly the bottom one. A split seam, a bulge or a popped crimp is the usual damage. It is a cheap part to replace and it will keep happening until the water stops standing there.
Reducing it
Clear the trough and the downspouts in the fall, so there is no debris holding water.
Make sure the discharge is clear and the extension slopes away, so water does not stand in the bottom.
Remove or redirect an extension that discharges onto a surface that ices over.
Consider removing the bottom section over winter on a problem downspout, letting it discharge higher and further out — which looks odd and works.
Heat cable in the trough and down into the downspout, which is a legitimate measure on a chronic north-facing run.
And deal with the attic heat loss, because less meltwater means less to freeze.
If it has already split
Replace the damaged section rather than taping it, which is a cheap part.
Check the whole run, because an ice column splits more than one joint.
Check the trough above it, which will have been backing up.
Check the fascia and the wall below, because a split downspout has been running water down the wall all winter.
Check the foundation and the basement at that corner, which is where the consequence shows.
And do the fall cleaning and the attic work before next winter, because the same conditions produce the same split.
Common questions
Why does my downspout split in winter?
Water standing in the bottom freezes first, the column builds up, and expansion splits the seam.
Can I remove the bottom section for winter?
On a chronic problem downspout it genuinely helps, as long as the discharge is still clear of the foundation.
Where does it usually split?
At an elbow, most often the bottom one.