SPVs are typically used in MBS issuance to get the loans off the issuing bank's balance sheet, freeing up that balance sheet space to make more loans and providing bankruptcy-remoteness (i.e., the SPV would continue to function even if the issuer went bankrupt) to investors. This is why a securitizing bank would transfer loans to an SPV.
However, given that you're describing an SPV sitting between an originator and an issuer, what you're describing might be a warehouse facility, which is a way of cheaply funding loans (in an incredibly unstable way) while the issuer puts together loan pools for issuance.
In general, the main motivators of SPV use are legal reasons (ensuring clear treatment of assets under all circumstances) and cost-minimization (bank balance sheet funding is relatively expensive as compared to off-balance sheet funding).
For a full treatment of SPVs in securitization, see Gorton and Souleles. While the text focuses on credit card securitization, the principles described apply equally to MBS.
Edit: I also endorse BKay's answer, which gets at some of the economic concepts motivating SPV creation.