Essentially it is due to the sugar, ribose, that is part of the RNA structure. This is chemically distinct from the deoxyribose that constitutes DNA. In ribose, there is a free hydroxyl group attached to the pentose ring, that makes the RNA molecule prone to hydrolysis, and hence more unstable. In essence, this means RNA will react with water and breakdown. Deoxyribose does not have this free hydroxyl group at the 2' position and it therefore more stable.

Not an expert, but I also think the use of uracil in RNA also is a factor in its stability. Uracil is a waste product of cytosine deamination, and the cell can get confused by whether the uracil is associated with cytosine breakdown or the RNA structure.

Finally, DNA has a phosphodiester backbone that helps protect the reactive nucleotides (they contain amino are carboxyl groups). Furthermore, the hydrogen bonds between the bases constitute the helix and further protect the structure. There are also other properties of DNA that add to its stability.

Well, I don't know if it is the chemistry that differentiates between RNA and DNA stability. Decades ago, I worked with RNA bacteriophage, and it was no problem isolating infectious RNA from infected bacteria provided you minimized [abolished] RNase activity during extraction procedures. My guess is that that is the major factor in the mRNA instability. Chelating agents effectively abolish DNase activity, but not RNase. RNase contamination of of chemicals [e.g., sucrose] in the old days was a real headache. Hope this helps.