To give further context for this quote, it comes from a script he used during a time when he had a 15 minute radio show on the WNEW radio station in New York in 1944-1945 for a 12 week period. Ed Cray's book Ramblin' Man provides more context around the quote:

I hate a song that makes you think you’re not any good. I hate a song makes you think you are just born to lose. Bound to lose. No good to nobody. No good for nothing. . . . Songs that run you down or songs that poke fun at you on account of your bad luck or your hard traveling.

I am out to fight those kind of songs to my very last breath of air and my very last drop of blood.

I am out to sing songs that’ll prove to you that this is your world and that if it’s hit you pretty hard and knocked you for a dozen loops, no matter how hard it’s run you down or rolled over you, no matter what color, what size you are, how you are built, I am out to sing the songs that make you take pride in yourself and in your work.

In full, the quote is clearly something of a mission statement for his WNEW radio show, where he talked and sang songs. It's probably fair enough to say that 'I am out to sing songs that'll prove to you that this is your world' is likely a reference to his most famous song 'This Land Is Your Land'.

At this stage - 1944-1945 - Guthrie was 32-33 years old, and by this stage had made a name for himself both in California in the late 1930s and in New York in the early 1940s, where he was championed by the ubiquitous-promoter-of-anything-folk Alan Lomax (who had recently turned Lead Belly into a sensation). So by 1944-45, he was established in New York folk music circles, had put out the iconic Dust Bowl Ballads album, and had been recorded by Alan Lomax for the Library of Congress.

In 1945, Guthrie wrote the liner notes for an album, Folksay, that came out on Asch Recordings (and which he sang some songs on). In those liner notes, he expanded further on songs he didn't like:

Hollywood songs don't last. Broadway songs are sprayed with a hundred thousand dollars to get them going, and they last, we’ll say, a few months at most. The Monopoly on Music pays a few pet writers to go screwy trying to write and rewrite the same old notes under the same old formulas and the same old patterns. Every band on the radio sounds exactly alike. . . . Do the big bands and the orgasm gals sing a word about our real fighting history? Not a croak.

I would assume that the "Hollywood songs" and "Broadway songs" here are the songs that Guthrie is opposing with his "songs that’ll prove to you that this is your world" that sing about "real fighting history" that "that make you take pride in yourself and in your work". Guthrie is of course wrong about these songs not lasting - as a modern listener to music you'll most likely encounter the Hollywood and Broadway songs as 'the Great American Songbook' as interpreted by the likes of Frank Sinatra, Ella Fitzgerald, various jazz musicians, or more recently Michael Buble. This is songs like 'My Funny Valentine', 'Let's Call The Whole Thing off', 'Stardust', 'Let's Do it (Let's Fall In Love)'. These are the songs of the likes of Cole Porter, Rodgers and Hart, the Gershwins, and Irving Berlin - presumably these are some of the songwriters Guthrie would be thinking of with the line 'few pet writers'.

Guthrie is implicitly making a distinction between the capitalist music of Hollywood and Broadway (note that he focuses on the 'Monopoly on Music' - presumably the music publishers whose offices were congregated together on a street dubbed Tin Pan Alley) and the authentic music of the people - folk music. Guthrie in the 1940s in New York was certainly in the orbit of leftists like Pete Seeger (who was one of McCarthy's targets) and Alan Lomax (who had a long FBI file that stretched for decades because they were suspicious of his leftist and possibly communist tendencies) who - if not actually card carrying members of the Communist Party - were clearly interested in using folk music as a way to raise class consciousness amongst the working class. Guthrie claimed in 1949 that "the best thing that I did in 1936, though, was to sign up with the Communist Party", but this was not the truth; 1936 was a little before he got properly politically conscious, and in any case Guthrie was too much of a free spirit with his own ideas to be comfortable spouting the Communist doctrine of the era, and his politics were probably more democratic socialist - he also apparently never actually signed up as a member.

But while he might not have carried the membership card, Guthrie was definitely someone who saw his identity as a folk singer as being about the same working class that the Communists were aiming to the be party of. With that in mind, it's likely that the working class, for Guthrie, were fairly specifically the 'you' in the statement quoted in the title by OP. In making that statement about songs that make you feel that you are not any good, Guthrie is basically pointing at Hollywood and Broadway as being about glamour, glitz, fancy clothes and finery - all of which costs money, which working class people didn't have. The working class, Guthrie, is arguing, did not see themselves reflected in Hollywood movies - and thus the songs in Hollywood movies - or in the plays of Broadway. So in some ways, the quote is something like a 1940s version of the 1990s complaint that there's no way that most of the characters in Friends could actually afford to live in an apartment that size in New York.

Take, for example, the 1937 Fred Astaire/Ginger Rogers vehicle Shall We Dance - the film that launched a few different George & Ira Gershwin songs into the Great American Songbook - in particular 'They All Laughed', 'Let's Call The Whole Thing Off' and 'They Can't Take That Away from Me'. The film is of course about a ballet dancer and a tap dancer - not exactly the most working class profession in the mind of a (self-styled) Okie cowboy. These songs are lyrically all light and fluffy, and follow a particular formula, and there's a certain intended refinement in the lyrics and vocals. In contrast, a Guthrie ballad like 'Pretty Boy Floyd' tells the story of a Robin Hood figure in Oklahoma who robs from the rich and gives to the poor. 'Pretty Boy Floyd' ends with the lyric:

You won't never see an outlaw

Drive a family from their home.

Which is very clearly meant to be contrasted with the banks and landlords who had little mercy during the Dust Bowl years in Oklahoma (a severe drought during the mid-1930s). Guthrie is implicitly contrasting the likes of 'Pretty Boy Floyd' - folk songs with a message, which was Guthrie's big specialty - with, say, 'They All Laughed'; the kind of thing you'd see in a big and glitzy Hollywood production.

Additionally, as a good 1940s leftist, Guthrie was trying his best to be anti-racist - he moved in the same leftist New York circles, for example, as Abel Meeropol who wrote ‘Strange Fruit’ (though he certainly had a history of racism, having performed in blackface and used the N-word on air in California, which he stopped after receiving a letter from a disappointed Black fan). Nonetheless, the 1940s Guthrie saying ‘you’ in OP’s quote was implicitly including people of colour - and Hollywood musicals very often portrayed not only a rich and successful world but one that was very much white-dominated.

So, for Guthrie, these songs seemed to portray a world where the likes of Guthrie weren't invited, where everyone was rich and successful and pretty and sophisticated (and white), a world of people born to win. In his music - and on his radio show on WNEW where he played his songs - Guthrie was trying to justify why his music was not going to sound like the sophisticated orchestra-backed stylings of a Fred Astaire - instead, his relatively musically simple, and (immaculately carefully styled) rough-around-the-edges folk was that way because he wasn't trying to make music that would sound good while rich looking people who might be bankers and landlords danced - instead, he was making music that championed the working class, and that (self-consciously) reflected working class values, highlighting underdogs and celebrating people who would never get celebrated in Hollywood.

(My main source here on Guthrie is Ed Cray's excellent biography, Ramblin' Man: The Life And Times of Woody Guthrie.)