To answer the first part of your question: nobody really knows.
There are a number of suggested origins:
- Brass tacks is rhyming slang for facts[A,B,C]
- Reference to measuring points in a fabric store and precision[B,D,E,F,G]
- Reference to the tacks used in upholstery, typically on the underside of furniture[B,E]
No. 1 is disputed due to the apparent US origin of the phrase (first use and use of "get down to"), since rhyming slang isn't common in the US. It may just be a coincidence rather than the true origin of the phrase.
No. 2 supposedly refers to buying fabric in a store, and taking it to the brass tacks (the measurement device, made of brass tacks hammered into a surface at precise intervals to measure distance) to ensure you get a correct or accurate quantity. In this case, when you are taking the fabric to the measuring device, you are "getting to" where that device is, with "down" possibly referencing the lower height of the measuring device (perhaps as compared to rolls of fabric, which are often stacked high):
4.1 [no object, with adverbial] Move or come into a specified position,
situation, or state:
"she got into the car"
"Henry got to his feet"
"you don't want to get into debt"
1. Towards or in a lower place or position, especially to or on the ground
or another surface:
"she looked down"
"the sun started to go down"
"he put his glass down"
"he swung the axe to chop down the tree"
1.4 To or at a place perceived as lower (often expressing casualness or
lack of hurry):
"I'd rather be down at the villa"
"I'm going down to the pub"
No. 3 may refer to reupholstering furniture, as to do that you would need to take out the brass tacks holding in the fabric, and with those tacks being on the underside you would need to physically move down under the furniture ("get down to"). It may also be similar to (get to) the bare bones of (sth.), since the tacks are part of the chair's construction rather than aesthetic additions (like the outer fabric on the chair). Arguments against this include that the phrase appears to date to the 1860s, whilst brass tacks have been used in upholstery for far longer.
Both No. 2 and 3 are supported by the synonymous phrase get down to brass nails,[F,H,J] since tack is another word for a nail:
A small, sharp broad-headed nail:
"tacks held the remaining rags of carpet to the floor"
To answer the second part of your question, the origin of the "get down to" part is also uncertain. In what seems to be the first usage of the phrase, it is written as "come down to" instead:
"No one we apprehend will accuse Washington of a want of patriotism, unless he
have some other object than the truth in view. For doing what he did, we, with
others, accuse people of selfishness, but we should in justice add that when
you come down to "brass tacks"—if we may be allowed the
expression—everybody is governed by selfishness, and if the merchant,
who refuses to take what is due him at 50 cents on the dollar, is selfish, the
debtor who insists on his doing so, is just as selfish."
— "Brass Tacks", E. H. Cushing (The Tri-Weekly Telegraph),
vol. 28, no. 133, ed. 1, pg. 2, para. 5
The piece doesn't explain the reasoning for using the phrase, but goes on to use it several times in a discussion of economy and currency:
"If the present currency ('brass tacks' again) is worth but two for one to
buy gold, food clothing, houses, lands, negroes, sheep and cattle, it is in
other words worth but 50 cents on the dollar, and will itself be bought, sold
used to pay debts with or loaned out at that rate. We do not say what ought
to be done, and what patriotism would seem to require. We are simply speaking
of what is and will be until human nature becomes altogether another thing
from what it now is. Who but a financial fool or a gambler or a speculator
(patriotism is out of the question) would borrow currency now to be repaid
a year hence in legal tender? What then? Why (down to the brass tacks)
borrow at the current exchange rate and pay again at the exchange current at
the time of payment."
— para. 6
"We urge the merchants to take the money through motives of patriotism and
fund it. We urge the planters to fund it likewise. We urge everybody to show
their patriotism practically instead of trying to get everybody else to do it
for their individual aggrandizement. We ask the people generally to come
down to "brass tacks." If they will theorize, act on their theories, and if
they will not act on their own theories, let them not lay themselves open to
the charge of hypocrisy and dishonest greed, by demanding of others what they
will not grant themselves."
— para. 11
Note that this piece was published in 1863, and so is now in the public domain (and so it is legal for large portions of the text, or the complete text itself, to be reproduced).