Insectivorous birds in particular often migrate over long distances. In summer they find many different kinds of insects in Northern Europe, but in winter insects are hard to find here. A good example are swallows. In summer they catch a lot of flying insects here, but in winter they have to go to more southerly places, due to a lack of flying insects.
So each winter 'our' barn swallows migrateall the way to Cape Town, South Africa! This is more or less also the case for many types of waders. They also love insects, supplemented by worms and crustaceans. That food is also scarce in Northwest Europe during the winter months.
The other extreme is formed by short distance migrators, such as the robin. This species moves from - for example - Denmark to the Netherlands in winter and finds the Dutch winters mild enough to survive. The robin in your garden is most likely a different bird in the winter than the one you saw in the summer!