תהלים פרק קיד, א בְצֵאת יִׁשְרָאֵל מִׁמִׁצְרָיִׁם בֵית יַעֲקֹב מֵעַם לֹעֵז: (הלל)
According to most commentators the words מעם לעז (referring to the Egyptians) indicates a people that does not speak the Holy Language. And this seems to be the intent of Rashi throughout his comments on the Chumash and the Gemara, when he translates a word from Hebrew to another language he writes "בלע"ז", which is an abbreviation for "בלשון עם זר", indicating not the Holy Language.
But this explanation does not seem to assist us in understanding this verse, for what is the connection between the Jews and another nation that does not speak the Holy language. If the verse is attempting to disgrace the Egyptians since they do not speak Hebrew, what is the disgrace since all the other nations of the world also do not speak Hebrew, and none of them are called by this term?
A weak answer would be that the verse is attempting to reinforce the praise of the Jews that they continued to speak Hebrew while enslaved in Egypt and did not change their language. But why praise the Jews with this item, when the same Medrosh also mentions they did not change their pattern of clothing or their names.
Perhaps, the word לעז has a different meaning. The Gemara frequently uses the phrase מוציא לעז which implies gossip and perverse speech. Thus, it would be possible that this verse from Tehillim implies that the Jews went out specifically from a place where impure speech was the norm.
In order to fully understand this connection, one needs to explain another verse found in Parsha Ki Savo (26, 6) וירעו אותנו המצרים ויענונו ויתנו עלינו עבודה קשה. In many other places in the Torah, when describing how the Jews were treated by the Egyptians, the verses use לי, לנו, לעם. All of those verses use a relationship ל, and not the connection את that is found in Devarim אותנו.
One can explain that the וירעו אותנו alludes to the Aggadah that the Egyptians were embarrassed in front of the civilized world that they were afflicting the Jews, a quiet and harmless nation, that did not seem to warrant the hatred of the Egyptians. Therefore, the Egyptians sought to justify their behavior to the world by slandering the Jews whenever possible, taking actions out of context, and applying higher than normal standards to the Jews (similar to what we find today).
Now one can understand the verse in Devarim. The Egyptians first made the Jews appear to be bad and evil and then, with apparent justification, they afflicted the Jews.
Now, one can understand the verse in Hallel that describes the Jews leaving a land where the people slander and viciously disgrace the truth.
