Question

Explain how the three main groups of protists give
rise to the three major high groups of fungi, plants and animals.

Answer 1

This is an interesting and difficult question to answer for several reasons -
1) The classification of the protists is still being debated, I believe. At the very least, it seems the older view of protists being seperated by whether they are animal-like or plant-like is inaccurate :D

This is a picture of one of the proposed trees of protist evolution (as you can see, rather messy): http://bioweb.uwlax.edu/bio203/2010/renner_brad/28_04EukPhylogeny.jpg

One method of organizing these various branches is to place these groups of protists into larger "supergroups." So, for example, the branch leading into your red algae, chlorophytes, charophyceans, and plants is collectively referred to as "Archaeplastida." The big branch leading to the Ophistokonts and Amoebozoa is collectively referred to as the "Unikonta."

2) The method by which unicellular organisms (ancestors of both protists and these higher kingdoms) evolved into fungi, plants, and animals is also still being researched, and I am uncertain that you'd find a definite answer.

However, there are interesting answers out there, and I suggest you try Googling the two supergroups I suggested above - reading about "Archaeplastida" may lend you insight on how plants developed, and reading about "Unikonta" and "Ophistokonts" will help you understand how fungi and animal developed.

Best of luck!