What does the word "respectively" mean in software development?

This is not really a programming question. It is always true in English that when you say "A and B are true for X and Y respectively," you mean that A is true for X and B is true for Y.

So to take your examples

C# Textreader is another way to read a file and TextWriter is another way to write a file...

I added a text box I called tbUpdate and button I called btUpdate...

In the right hand column, we’ll use three TextBlocks; the first (with the value Likeability) and third (with the value %) as fixed labels...

WorkerSupportsProgress which holds true/false values that lets the backgroundworker report progress and WorkerSupportsCancellation which holds true/false values that let the backgroundworker cancel an async progress...

"Respectively" is used when presenting two lists that have a direct correlation.

Those sentences can be rewritten like so:

C# Textreader, to read, and TextWriter, to write, are another way...

OR:

C# provides another way to read, with TextReader, and to write, with TextWriter, ...

Each item in both lists has a direct relation with respect to the corresponding item in the other list.

I added a text box and button I called tbUpdate and btUpdate respectively...

means

I added a text box I called tbUpdate, and a button I called btUpdate ...

Just an addition, my guidebook prefer the latter form because it's easier to read (imagine if there are 5 items being listed).