The process of DNA transcription can be split up into three stages: initiation, elongation and termination.
First, an enzyme called RNA polymerase binds to a certain sequence of bases called the promoter region that can be found in the beginning of a gene on the DNA molecule. When this happens, the DNA double helix is uncoiled and the antisense strand is exposed and ready to be used as a template for making a copy of the sense strand on which the gene is found. This is the initiation stage.
Then, free nucleoside triphosphates bind to complementary bases, just like in DNA replication. An important difference however, is that where T would normally have bound to A, U binds instead, since thymine is replaced by uracil in the mRNA. C and G complement eachother as normal. The RNA polymerase makes sure covalent bonds are formed between the nucleotides.
This is the elongation stage and it goes un until a special sequence of nucleotides on the antisense strand called the terminator is reached. In the termination stage, RNA polymerase detaches itself from the DNA molecule, the mRNA exits this little bubble that has formed, called the transcription bubble, and the DNA rewinds itself into the original double helix.