actually.. not quite
the reason for using an interface is not so much the software they can come with (which can be nice in some cases) but for using XLR style microphones, listening to your voice while you record live and being able to mix in multiple sources at once. many professional streamers are using such devices as most use xlr style microphones, listen to their voice live and may need to mix in or output to different devices.
many have a weak amplifier built in which normal headphones will work with but for higher power headphones you would want an amplifier connected on the output. similar to a pc, while your dt770 may work with it minus an amp it wouldnt be stellar without using an amplifier.
there are many cheap audio interfaces out there although i tend to recommend things like the scarlett solo which i know are quality products. good amplifiers to pair with it would be the schiit magni, o2, fiio a3 in that order. i've heard of some quality issues on the cheaper ones which is why i dont generally list them but for on the cheap they are fine if you are okay with some quality issues.
if you do not care about listening to your voice while you record (you can do this without an interface but may have a micro lag delay through windows), do not care about mixing or native XLR input (you can use phantom power adapters and xlr-3.5mm adapters instead) or just plan to use a usb mic (interfaces do not work with usb) you really dont need an interface and a decent soundcard would do (or using onboard+amp, soundcard+amp, external dac+amp, etc).
as far as just amplifiers, the schiit magni will power just about anything on the market and could be used for a long time through any of your future upgrades. the o2 is not far off either and is a quality unit but more expensive. for on the cheap the fiio a3 is not bad but given its portable i've heard of noise while charging and its not going to handle every headphone on the market well.
explanation of some of the options you are left with:
onboard audio + external amp
cheap option, powerfull enough to drive your headphones, quality of the sound still relies on your onboard dac so may not be quite up to snuff or may have issues depending on how good your onboard audio is.
internal soundcard
cheap option, strong cards like the z may work although will still lack power to drive high power headphones effectively. you get virtual surround sound, software and some other options you may or may not need.
internal soundcard + external amp
mid level option, amp is strong enough to drive headphones yet if you use a soundcard like the dx you can have a good quality dac unit as well as software for virtual surround and aux outputs and inputs for things like speakers, analog mics, etc. generally the go to choice for people who want both virtual surround as a must have and need to power headphones a soundcard alone can not handle well. why the dx? it has a fairly weak amplifier yet the same dac as the $200 stx. in comparision the z has a stronger amplifier but weak dac.
external dac + external amp
this route is limited to stereo only (except using razer virtual surround only) but is certainly high quality. products range in price from fairly cheap (e10k, audioengine d1) to reasonably expensive but still budget priced in terms of audio (schiit magni + modi stack). expect quality audio out of such setups but you are limited to just one thing: quality stereo audio output for headphones. some (modi uber edition for example) adds aux outputs or optical inputs. if you use a usb microphone there is no real need to worry about having a soundcard or interface.
so, what is best for you? that depends..
A) do you plan to use an xlr microphone?
B) do you need to live monitor your voice?
C) do you need virtual surround sound?
D) do you need aux input/outputs
for on the cheap, an e10k would likely work (ext dac+amp combo box). if you want virtual, the xonar dx + schiit magni (or a3).
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the lepai 2020 amp is meant for speakers not headphones. plug headphones into a speaker amp and you will likely fry them if you turned the volume up. headphone amplifiers are marketed as headphone amplifiers.