The literal answer to your question 'what determines the price of an ETF' is 'the market'; it is whatever price a buyer is willing to pay and a seller is willing to accept. But if the market price of an ETF share deviates significantly from its NAV, the per-share market value of the securities in its portfolio, then an Authorized Participant can make an arbitrage profit by a transaction (creation or redemption) that pushes the market price toward NAV. Thus as long as the markets are operating and the APs don't vanish in a puff of smoke we can expect price will track NAV.
That reduces your question to: why does NAV = market value of the holdings underlying a bond ETF share decrease when the market interest rate rises?
Let's consider an example. I'll use US Treasuries because they have very active markets, are treated as risk-free (although that can be debated), and excluding special cases like TIPS and strips are almost perfectly fungible. And I use round numbers for convenience.
Let's assume the current market interest rate is 2% and 'Spindoctor 10-year Treasury Fund' opens for business with $100m invested (via APs) in 10-year T-notes with 2% coupon at par and 1m shares issued that are worth $100 each. Now assume the interest rate goes up to 3% (this is an example NOT A PREDICTION); no one wants to pay par for a 2% bond when they can get 3% elsewhere, so its value goes down to about 0.9 of par (not exactly due to the way the arithmetic works but close enough) and Spindoctor shares similarly slide to $90. At this price an investor gets slightly over 2% (coupon*face/basis) plus approximately 1% amortized capital gain (slightly less due to time value) per year so it's competitive with a 3% coupon at par.
As you say new bonds are available that pay 3%. But our fund doesn't hold them; we hold old bonds with a face value of $100m but a market value of only $90m. If we sell those bonds now and buy 3% bonds to (try to) replace them, we only get $90m par value of 3% bonds, so now our fund is paying a competitive 3% but NAV is still only $90.
At the other extreme, say we hold the 2% bonds to maturity, paying out only 2% interest but letting our NAV increase as the remaining term (duration) and thus discount of the bonds decreases -- assuming the market interest rate doesn't change again, which for 10 years is probably unrealistic (ignoring 2009-2016!). At the end of 10 years the 2% bonds are redeemed at par and our NAV is back to $100 -- but from the investor's point of view they've forgone $10 in interest they could have received from an alternative investment over those 10 years, which is effectively an additional investment, so the original share price of $90 was correct.