If I take [から] out of the sentence, the meaning is still the same.
Not exactly. In translation, both から and それで will take on the meaning of 'because' in the English translation. This gives the effect of から appearing to have no real function here. However, それで is a weaker form of 'because,' and から in this sentence relies on the wider context of the preceding sentences. Therefore, if you were to remove から from the sentence in question, you not only loose the strength of the cause/effect association, but you also what the real cause and effect are.
In effect, we're looking at multiple causations in this sentence.
それで as defined by Jisho.org means
and; thereupon; because of that. Other sources also include the meaning
then; so; as a result; that’s why. I personally like to think of それで as the word それ followed by the particle で.
The で particle has several functions, but one of them is to express reason or causation. Therefore, when you attach the particle で to それ, you get the literal meaning of
that/it causes. In the context of the sentence you've included, my personal take on a literal translation would be 'by reason of that situation/motivation.'
In short, それで explains what follows as happening by reason or cause of what preceded it. Hence, in English, we would typically translate it as meaning
because of that.
から, on the other hand is a hard 'because,' establishing a stronger sense of causation than それで. からです is the Japanese equivalent of the English
it is because.
Lets break down the sentence and see if we can make sense of all these moving parts:
This will take context to the two other sentences, though I have not included them:
それ: This points to the second sentence, or Brad's commitment/decision to save money.
で: marks reason/cause
来年: next year
移ることができる: able to move
からです: it is because - 'it' in this case actually refers to the first sentence saying that other apartment is expensive. Note that this particular usage of から relies heavily on the context of what has been said before this sentence.
Bringing it all together, I get the following (liberal) translation including context instead of
That apartment is expensive because Brad's commitment to save money has made it so that he is able to move.
Shortening the sentence:
It's expensive because his commitment has made it so that he is able to move.
それで in this sentence actually comes into play in the phrase
...his commitment has made it so that... while から in this sentence comes into play with the phrase
It's expensive because... Therefore, this sentence actually has two causation relationships happening, and needs both それで and から to say what it does.