Step 1: CpG islands are regions with a high frequency of CG dinucleotides and are often located in gene promoters.
Step 2: Methylation of CpG islands at the promoter region can inhibit gene transcription by several mechanisms:
- Preventing RNA polymerase binding (A) as methylation physically blocks the binding of RNA polymerase to the DNA.
- Facilitating repressor binding (B) where methylation attracts repressor proteins that further inhibit transcription.
- Facilitating heterochromatin formation (C) which leads to a more compact and transcriptionally inactive chromatin structure.
- Inducing euchromatin formation (D) is incorrect because methylation generally leads to gene silencing, not activation.
Final Answer: Methylation inhibits transcription by preventing RNA polymerase binding, facilitating repressor binding, and facilitating heterochromatin formation.
\[\boxed{\text{Correct Answer: (A), (B), and (C)}}\]