The expression ‘his language often betrays him into dullness,coldness,or worst of all, rudeness’ means that

In the past, learning English as a separate subject seemed relatively easy. The textbook selected and graded items of language which were put into context and then practised intensively. New items were carefully controlled so that the student could cope quite easily. Now that English is used as a medium of instruction, however, all this has changed. Unknown items of grammar and vocabulary appear in texts which attempt to explain new and often difficult information. Difficulties with the language interact with difficulties as regards the subject matter. The student’s reading in his own subject slows down and his comprehension becomes less secure. He expresses himself slowly and often fails to convey his ideas exactly. He is disappointed to find that under pressure he makes a lot of unnecessary mistakes in areas where he knows the correct language forms. His social relations are difficult as he cannot find the right phrase quickly enough to keep a conversation going; so his language often betrays him into dullness, coldness, or worst of all rudeness. Instead of the students being in control of the language, the language seems now to be in control of the student.
All of this can be very depressing and the student can start to feel very anxious. Working in a foreign language is also very tiring, and the concentration and self-discipline required to correct one’s mistakes is very great indeed.