From what i gather. Arsenal offered to pay 60% of Sahin's wages and wanted an option-to-buy-clause for £14m. But Mourinho wouldn't agree to the option clause and stalled causing Wenger to bail on the deal.
Then the Scousers swooped in and offered to pay 70% of his wages. However, there are conflicting reports on whether they got the £14m option clause or not. I'm guessing probably not. Then again Mourinho might've done his old pupil a favour and offered him the clause.
Being financially outbid isn't that shocking to any Arsenal fan. Happens all the time. David Silva, Juan Mata, Eden Hazard, Ashley Young, Hugo Lloris, Nuri Sahin, Yann M'Villa etc. All linked with Arsenal, all ended/will end up in other clubs.