Why did Germans migrate towards the East?

Since Medieval period many monarchs in Central and Eastern Europe invited German settlers to settle in their (sometime) newly build city. This include kingdom of Hungary which invited the Germans to Transylvania and kingdom of Poland which invited them to Krakow and Danzig. The Germans were obvious choice due to high population and their experience in city life. In fact many cities in Central and Eastern Europe adapted laws (such as Magdeburg laws) based on Germanic city law as their governance law.

And the Germans also came uninvited too. Many Germans, especially second or later born sons of nobility joined Christian military orders. These orders, such as Livonian Orders and the famous (or infamous, depens on your view) Teutonic Orders then engaged in Crusade against pagan Baltic tribes in modern-day Latvia, Lithuania and Estonia. These settlers were the predecessors of Baltic Germans population.

Apart from these we also had Hanseatic traders of North Germanic city states, such as Lubeck and Mecklenburg, who ventured to eastern Baltic Sea, as far as Novgorod in Russia, for pursuit of commerce. They traded grains from Poland and fur from Novgorod.

Why, for example in this photo, were they refered to as colonists? Surely in 19th century Europe the government of (in this case Russia) would have been able to stop an influx of Germans?

Actually Russia used to encourage Germans settlement. Under Catherine the Great (who herself was a German-born princess) Russia conquered large part of what is now southern Ukraine from the Crimean khanate. This newly acquired area, together with the still sparsely populated lower Volga, need settlers so that Russian rule there could be secure and profitable, so the Germans were invited as settlers.