Explain what the archaic word "trow" means ?

The word "trow" meant something like "think" or "believe."
You can find this word used in old literature fairly often.  Shakespeare used the word
on a few occasions.  For example, see this passage from The Taming of the
Shrew. (This is from the start of Act I, Scene
2.)

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My best beloved and approved friend,

Hortensio; and I trow this is his house.
Here, sirrah Grumio,
knock, I say.

What Petruchio
is saying here is that he thinks or believes that this house belongs to his friend. 
This would make sense if you put one of those words in there.  Petruchio would be saying
"I believe this is his house."

So, "trow" simply means to
believe or to think that something is the case.