Why is Travis a "walking contradiction"?

Travis is indeed a character of walking contradiction.

He had a hatred for sex and pornography and yet he spent much of his time in some porn theatres. During the course of the movie, however, he tried to

dissuade Iris from continuing in prostitution. ¹

He was very much infuriated with the surrounding corruption and crime and yet he started to work for an illegal gun dealer.

"He wanted to be good to his body, yet he constantly took pills and poured schnapps on his breakfast." ²

Betsy, however, might not have been aware of this trait of Travis as she knew him for only a brief period of time.

The point is that she was quoting the lyrics of The Silver Tongued Devil and I and might wanted to specially quote the following:

See him wasted on the sidewalk in his jacket and his jeans, wearin' yesterday's misfortunes like a smile.

which was in line for the appropriate depiction Travis' physique. However, she made a hotch-potch and left some words out of they lyrics and quoted in haste the phrase "walking contradiction".

So, in a nutshell, this can be considered as a coincidence that the song was reflecting quite accurately the personality of Travis, albeit being quoted from the lyric unknowingly by Betsy.

This is explicitly thoroughly summed up here in this SparkNote analysis of the quote:

Betsy quotes the lyrics to the song correctly but leaves some words out, and she mixes up their order, so that the "walking contradiction" phrase, which is most important to her, comes last.

[...] Travis is a walking contradiction. He is disgusted by sex and by prostitutes, yet he surrounds himself with pornography and takes prostitutes around in his cab. He wants to be good to his body, yet he constantly takes pills and pours schnapps on his breakfast. Betsy, of course, knows nothing about these quirks. She knows Travis only from the limited and awkward conversation they've had at the diner. Perhaps she is reminded of the song because of its first two lines, which she does not quote: "See him wasted on the sidewalk in his jacket and his jeans, wearin' yesterday's misfortunes like a smile." These lyrics describe Travis's physical appearance perfectly. The Kristofferson lyrics provide an unusual glimpse into Travis's character, though Betsy is not even aware of the elements in Travis's life that make them particularly true.

At the end, it is also noteworthy to mention from the same source:

[...] One might suspect that Schrader, the screenwriter, wanted this line to explain Travis to the audience.