You'll notice that all of these words include ch in German and gh in English. These originally represented the same sound: a voiceless velar fricative, written as
/x/ in the International Phonetic Alphabet. (It's the final sound in loch, if you pronounce that different from lock.)
Both English and German inherited this consonant from Proto-Germanic, where it's written
*h. (The star is because Proto-Germanic itself was never written down, it's a scholarly reconstruction:
h is just the letter scholars use when talking about it.) It often came from Proto-Indo-European
*k through Grimm's Law, though there were other sources as well.
In German, that sound was written ch, and has survived pretty much untouched until modern times. In English, however, the sound was written gh. And somewhere between Chaucer and Shakespeare, it disappeared entirely: this is called H-loss or the taught-taut merger.
P.S. Different dialects did different things with the
/x/ when it disappeared. Some turned it into
/f/ after back vowels, which is how we get laugh and tough. Others lengthened a preceding vowel, which is why light has a "long i" sound. Scots still preserves
/x/ even to the modern day.