aebe jeopsokhajamaja janaegi baro boyeoseo songgeumhagi swiwosseoyo.

-자마자 means as soon as. It attaches to a verb stem and shows that the next action or situation happens immediately after the first one.

So: 접속하다 = to access / log into
접속하자마자 = as soon as (someone) accessed / logged into

In this sentence, it means the balance appeared immediately after logging into the app.

Because 접속하다 normally uses 에 to mark the place, service, or network you connect to.

So: 앱에 접속하다 = to access / log into the app
사이트에 접속하다 = to access a website
인터넷에 접속하다 = to connect to the internet

Using 을/를 with 접속하다 would sound unnatural in standard Korean.

Here, 보이다 is being used, not 보다.

보이다 can mean to be visible / to appear / to be seen.
When something is visible, that thing is treated as the subject, so it takes 이/가.

So: 잔액이 보여서 = the balance was visible / the balance showed up

If you used 보다, then you would say: 잔액을 봐서 = because I looked at the balance

That would change the meaning. The original sentence focuses on the balance being displayed clearly, not on the speaker actively looking at it.