How do I connect my washing machine's waste pipe to a small connector?
That looks like a connector used to lengthen the waste pipe. It is designed to accept the common waste pipe sizes and most waste pipes have a flexible end - visible on the waste pipe on the back of the machine.
Once you have checked that that pipe goes into a suitable drain (with U bend) and it is not too long (6 feet is a usual max, but do read the machine instructions), and that extension is not blocked then it should work fine.
Based on the comment:
if you want the security of adding hose clamps, then do not make them too tight. I would make sure to push the pipe fully deep over the barbs or ridges - you can see the bulge caused by the ridges on the other side already.