How to load the program? (UART)

OK. For TI AMxxxx serial SoC, the memory addressing space after 0x80000000 is used for DRAM (DDR2, 3 ...). The CPU has DRAM controller but it needs to be initialized so that it can know the external DRAM configuration. And the initialisation is usually done in the boot loader.

Since you are newbie, you may now know the boot process very clear. Here is the basic concept.
Power up --> ROM boot loader (RBL) --> User boot loader (UBL) --> Main program

After powering up, RBL will try to load UBL from some sources depends on some pinout settings at boot. For example, the external SPI NOR flash, SD card, UART port, USB port, Ethernet, or internal flash memory. This will vary depends on SoC to SoC model. For AM437x, you can look at TRM page 175 for details.

If a valid UBL image is found then RBL will load it into internal SRAM and jump to UBL's entry point. Now, UBL is running. UBL is designed by you or someone else. UBL should initialise necessary peripherals such as DRAM and then load main program from external SPI NOR flash or SD card... to DRAM. After that, it should jump to main program's entry point. Now, your main program will get executed.

Please note that RBL is also called "ROM Code". There is a famous UBL program called U-Boot which can be used to load Linux kernel to DRAM. So, if you are using Linux then you could reply on U-Boot. But it is a little "fat" and I don't like it at all. I prefer to design my own UBL.

If you are not using U-Boot and Linux. You have a simple program and just want to get it run on target. OK, you can load it into internal SRAM. According to datasheet, there is 256KB SRAM on-chip. You can load a small program to SRAM by JTAG debugger. For this to work, you also have to change the memory map definition of the linker command file.