Next, the sentence states that they took "our" jobs. So, taking this sentence literally, we may conclude that the immigrants are doing exactly the same job "we" did; they simply replaced us at our jobs. This signals an implied judgment. The judgment is that, if there is any difference in the way immigrants perform on the job compared to the domestic population, this difference is irrelevant, completely unimportant. This judgment implies that, whoever uses this sentence thinks immigrants are completely unnecessary and that they are not bringing anything new and valuable to the employers. According to the sentence, the same jobs would have been done by the domestic population anyway.
The logical reality is, however, different. If immigrants are really taking something the domestic workers rightfully own (their jobs) that implies that the domestic employers have a moral obligation to employ only the domestic workers. If this was true, this would be a coercive relationship in which the employers are forced to stay in the relationship even when they don't want to. Second, we can logically deduce that immigrants must be providing some additional benefits to the employers; they are providing better quality of service or working at a lower wage or discovering new kinds of jobs that make the old jobs obsolete. Otherwise, how would the immigrants persuade the employers to employ them if the same jobs can be performed equally well by the domestic workers? The employers think these benefits are large enough to justify going through the costly process of starting a relationship with a new employee and perhaps ending a relationship with an old one.
But, this is not the story that the sentence "Immigrants take our jobs" says. That sentence, in fact, doesn't say anything truthful about immigration, but it says much about the attitudes and beliefs of the person that uses the sentence.