|You don't have to get tangled up in|
the sentence. Just separate each part.
"Winners in a party primary have a financial advantage because once they are selected by the voters as the party candidate, the state party will use its money to support their campaign."
If you break this sentence into pieces, each piece is clear:
(1) Winners in a party primary have a financial advantage
(2) because [=the reason they have an advantage is that] once they are selected [upon being selected/after they are selected] by the voters as [=to be] the party candidate,
(3) the state party will use its money to support their campaign.
Each part of the sentence is called a separate "clause." To say the same thing more simply would take more words:
"Winners in a party primary get an advantage. The advantage is that there is more money for their campaign. The political party will give them the money they need. Until the voters choose the candidates in the primary election, they cannot get money from the party. The candidates have to raise money from private contributors or supporters."
I hope this explanation helps. You can always ask questions. I check the blog regularly and will be happy to answer. If I don't have an answer, I will find someone who does.
There are two more parts coming about the mid-term elections: candidate messages, and why it matters. I hope you find them of interest.