In a nutshell, the TEE does have its own bootloader.
The basic idea is that the CPU boots from ROM. This ROM is programmed to load a software image from persistent storage, verify that it is signed by a public key which is stored in the ROM, and transfer the execution to this software image. On an ARM platform with TrustZone, the CPU starts in secure mode, so this first software — the TEE — is executed in secure mode. At some point the TEE switches to non-secure mode and loads another software image: the “rich OS”, i.e. the main OS running on the platform.
The MMU with TrustZone separates a secure world running the TEE and a non-secure world running the rich OS. The non-secure world is less privileged than the secure world, so it cannot access the secure world's memory, in the same way a userland program cannot access the kernel's memory (though the technical details are somewhat different).
There isn't much publicly available documentation about TrustZone and associated technologies, but you can find some on ARM's website.
As real life is never so simple, what happens in practice is a chain where the ROM loads a first bootloader which loads a second bootloader, etc. As long as each stage verifies the signature of the code image of the next stage, only authenticated code can be executed. One of these loads a non-secure, non-authenticated bootloader in the normal world which in turn loads the main operating system (e.g. U-Boot and Linux). This is a secure boot chain, similar to what Microsoft requires for Windows 8. A secure boot chain ensures that only code approved by the hardware manufacturer can run. The idea of the TEE separation is to apply secure boot to the TEE, but not necessarily to the main OS.
Note that what secure boot guarantees is that only authenticated code runs. Malicious code can run if it got signed by the hardware manufacturer for whatever reason.
A Microsoft paper §6.7 gives an overview of an example secure boot chain. A Texas Instruments whitepaper describes a secure boot chain in more detail.