As a general rule you can't export the songs that are built into a Yamaha keyboard. There are usually functions to control which types of information will be transmitted by the keyboard via MIDI-- e.g., Keyboard Out, Style Out, and Song Out-- although I don't see anything like that in the PSR-280 Owner's Manual. The Owner's Manual for a keyboard that does have those types of functions will say that the built-on songs can't be transmitted via MIDI-- if you turn on the Song Out function, play back one of the built-in songs, and monitor the keyboard's MIDI output, you'll see that the built-in song isn't being transmitted. Since the PSR-280's manual doesn't list any functions for controlling which types of information will be transmitted via MIDI, you can try to transmit a song and see whether it works, but don't be surprised if it doesn't.
"DAW" stands for "digital audio workstation," and DAW software are programs that let you create, edit, and play back audio recordings on your computer. Most DAWs are commercial programs and can be rather expensive depending on how many virtual instruments, effects, and other features are included. However, there are some free DAWs, as well as some inexpensive commercial DAWs, plus "lite" versions of some commercial DAWs which are often bundled (i.e., included "free") with audio or MIDI equipment. There are also similar types of programs which might not be called a "DAW" per se-- e.g., they might be called a "MIDI tracker," "MIDI sequencer," or "MIDI editor." Some free programs that I know of are listed below:
Aria Maestosa
Avid Pro Tools First
LMMS
Ohm Studio
PreSonus Studio One Prime
Sony ACID Xpress
Tracktion T4
Willow Software Anvil Studio
Zynewave Podium Free
However, it is important to realize that even if the PSR-280 will allow you to transmit its built-in songs via MIDI as they're being played back, you'll probably need to edit the resulting MIDI data to make it compatible with another keyboard-- especially one from another company. Different manufacturers use different Bank Select and Program Change values for the sounds on their keyboards, as well as different System Exclusive messages. Yamaha keyboards generally have two types of voices-- GM or XG voices, and panel voices. GM voices are "universally available" on all GM-compliant keyboards, so a MIDI song file that uses only GM voices and other GM messages will work on any GM-compatible keyboard, although the voices might sound a bit different. But when a manufacturer includes built-in songs and styles on one of their keyboards, they usually choose the best-sounding voices so the songs and styles sound as impressive as possible. Unfortunately, the GM voices on a given keyboard usually don't sound as good as either the XG voices or the panel voices, hence you'll need to do some "revoicing"-- i.e., editing the Bank Select and Program Change values in the MIDI file to match the voices which are available on the other keyboard.