Hålla tummarna literally means 'to hold one's thumbs', and it's used to wish someone luck. So if you tell someone you're holding your thumbs, it's the same as saying “I'm crossing my fingers for you!” or simply “I'm wishing you luck!” in English.
You can follow the phrase with för and a noun, or att and a verb, for example jag håller tummarna för min vän (I'm keeping my fingers crossed for my friend) or jag håller tummarna att det ska hända (I'm crossing my fingers that it happens).
So why thumbs? This is an example of an interesting linguistic divide. In England, as well as many parts of Europe (France, Spain, and Sweden’s neighbours Norway and Denmark), crossing one’s fingers is the sign for wishing luck.
As well as the Swedes, the Finns, Czechs and Germans are among the nationalities who hold their thumbs instead.
Crossing one’s fingers is an old Christian symbol, and centuries ago people would form a cross with the index fingers of each hand, as a way of warding off bad luck. But holding one’s thumb is an older gesture, believed to date back at least as far as the Roman Empire.
It dates back to the idea of destiny and luck (good or bad) being something you could physically grasp. In this case, the act of holding your thumb symbolizes catching and holding an unlucky spirit or demon, to prevent bad luck.
In Sweden, when you hålla tummarna you might make the gesture of tucking your thumb into a closed fist, but Swedish-speaking Finns would usually make a thumbs-up gesture, with a closed fist and thumb pointing upwards, even though the phrase means the same thing.
Two final linguistic points: you might notice that hålla tummarna literally means 'to hold the thumbs', and there's no possessive pronoun, or in other words, there's no mina (my). In Swedish, when you're talking about body parts, it's much less common to use possessive pronouns in contexts where it's fairly obvious you'd be talking about yourself.
And the word tumme (thumb) itself is related to the English translation as well as to the German der Daumen. They come from a Germanic root word which originally meant 'thick' or 'fat', and it refers to the size of the digit compared to the other four fingers.
If you get confused by this phrase, don't forget that there's a simple alternative: you can just say lycka till! which means 'good luck'.
Jag ska hålla tummarna hela dagen!
I’ll be crossing my fingers/wishing you luck all day!
Nu måste vi alla hålla tummarna
Now we all have to cross our fingers/hope for the best