Normal is a sort of helper function that turns specialized structures into more simple representations, typically a List. TimeSeries is one of these special forms, but I wanted to use Select, and Select needs a List or at least something List-ish. Normal got me what I wanted. TimeSeries also has properties, so instead of Normal[dataTS] I could have used dataTS["DatePath"], and that's actually probably more natural.

For your other question, there is special syntax for Function. You can write an expression for the body of a function where arguments are represented by Slot expressions (which are typically just written as #) and & is added in post-fix position. So, Function[x, x^2] could be written as #^2&. In your case, I wanted to use Select with Between as the filtering predicate. Select expects the second argument to be a function (either a literal Function or a symbol/expression that can be applied to the individual elements). So, I needed to construct a function with a Between expression.

& is the punctuation to mark the end of a pure (or "anonymous") function. You can find the description of any keyboard shortcut by typing that shortcut into the search box in the documentation center. It's probably worth going through a tutorial on anonymous functions -- like Chapter 26 of the 3e of "An Elementary Introduction to the Wolfram Language".

In the details section of the TimeSeries help page, the docs note that the "Path" property generates the time-value pairs of the TimeSeries, and that Normal of a TimeSeries is an alternate way to generate those time-value pairs.

With

dataTS = FinancialData["GE", "Dividend", {{2020, 1, 1}, {2021, 12, 31}}]

you could do this

Select[Normal[dataTS], Between[#[[1]], {DateObject[{2020, 12, 1, 0, 0, 0}], DateObject[{2021, 9, 1, 0, 0, 0}]}] &]

With the legacy behavior, you could convert to DateObject in your selector:

dataLeg = FinancialData["GE", "Dividend", {{2020, 1, 1}, {2021, 12, 31}}, Method -> "Legacy"];
Select[dataLeg, Between[DateObject[#[[1]]], {DateObject[{2020, 12, 1, 0, 0, 0}], DateObject[{2021, 9, 1, 0, 0, 0}]}] &]