。
The feeling of being bound grew stronger and stronger.
Literal
Being-tied [quot-という] feeling [subj-が] intensified.
つながれている is the passive of つなぐ 'to tie / connect,' meaning 'to be tied / bound / connected.' The ~ているという is a relative-clause-like pattern where the clause X is flagged as a labeled feeling or thought modifying the noun: 'the feeling that one is being tied.' 募る (つのる) is the formal verb for emotions or tensions 'growing / intensifying / building,' most often in set phrases like 不安が募る 'anxiety mounts,' 思いが募る '(one's) feelings grow stronger.'