Why does "tu me manques beaucoup" mean "I miss you very much" ?

Short answer - because that's the way it is ! 

More useful answer - look ahead at the linked lessons, but don't be surprised if it is a struggle to get your head around ! (You will need to succeed with this in order to manage the use of the verb « plaire » as well)

Using manquer (à) to say you miss someone or something emotionally in French

Using manquer (de) to say you/something miss or lack in French

This is how I remember it, tu me manques = you are missed by me.  Similarly, il me faut = it is necessary that I, ca te plait =  it pleases you (you like it)