Tubocurarine is a non-depolarising neuromuscular blocking aqgent, that is, it acts as a competetive antagonist of Acetyl Choline at the Nicotinic receptors.(Toxicity/Effect: $-N$)
Atropine is a competitive antagonist of the muscarinic acetylcholine receptor, classified as an anticholinergic. (Effect: $-M$)
Neostigmine blocks acetylcholinesterase, increasing the concentration of ACh at, and stimulating both muscarinic and nicotinic receptors. (Effect: $+M+N$)
Hence, to counter the only nicotinic deficiency of ACh, we give Neostigmine, which will also stimulate the muscarinic receptors, leading to muscarinic side effects. To counter the latter, atropine is given. A similar formulation is used in a disease like Myasthenia Gravis, with a functionally similar lesion.