Non-Muslims say that they get answers from their idols or those whom they take as their gods when they supplicate to them; how can they identify that Allaah is the only God and that there is no god but Allaah?

All perfect praise be to Allah, The Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad, sallallahu ‘alayhi wa sallam, is His slave and Messenger. 

None answers the supplications except Allah, The Exalted. The idol-worshiper who claims that his idol answers his supplications should be reminded of the fact that his idol cannot ward off any harm inflicted on it, let alone benefit others. Your idol does not see or hear; how could it possibly answer your supplications? Allah, The Exalted, says (what means): {Indeed, those you (polytheists) call upon besides Allah are servants like you. So call upon them and let them respond to you, if you should be truthful. Do they have feet by which they walk? Or do they have hands by which they strike? Or do they have eyes by which they see? Or do they have ears by which they hear? Say (O Muhammad) "Call your 'partners' and then conspire against me and give me no respite.} [Quran 7:194-195]

This idol of yours did not cause you to exist, nor did it create you from nothing. All these facts can be easily grasped by the sound mind and reason. Allah, The True and Only God, is the One Who created you and caused you to exist, and He has full control over your affairs. Knowing that Allah, The Exalted, is the True and Only God can be realized by contemplating the signs of Allah in the universe and in His divine legislation. The greatness and meticulous creation of the universe proves that it has a Creator Who is Great, Omnipotent, and All-Knowing. This Omnipotent and All-Knowing Creator sent his noble Messenger with clear signs indicating his truthfulness as well as the fact that he was sent by that Creator, Exalted may He be. Whoever ponders over the Quran and contemplates its aspects of miraculousness will undoubtedly realize that it is from Allah, The Exalted, and that Allah alone is worthy of worship. There are many ways to attain the knowledge of the Oneness of Allah that the scholars mentioned. Shaykh As-Sa‘di said:

“Ways to know that there is none worthy of worship but Allah are many:

The first and the most superior of them is reflecting upon His names and attributes as well as His actions, which indicate His perfection, magnificence, and majesty. Indeed, this urges one to strive his utmost in worshiping Him as the True Deity and the Perfect Lord, to Whom belongs all praise, glory, majesty, and beauty.

Secondly, knowing that He is the One and only Creator of all things and the Disposer of all affairs; thus, you understand that He is the only one worthy of worship.

Thirdly, knowing that He alone is the bestower of blessings, apparent and hidden, religious and worldly. This compels the heart to be attached to Him, to love Him, and to single Him out alone in worship with no partners.

Fourthly, what we see and hear about the rewards for His allies, who duly celebrate His Oneness, like victory and immediate blessings, and His punishment of His enemies who associate partners with Him. This is a reason for knowing that He alone is worthy of all worship.

Fifthly, knowing the descriptions of those idols and false deities that are worshiped besides Allah and are taken as gods, as well as the fact that they are deficient in every aspect (while Allah, The All-Rich and Self-Sufficient, is not in need of anything). They cannot bring about for themselves or their worshipers any benefit, harm, death, life or resurrection. They can neither help nor avail those who worship them with the weight of a mustard seed in bringing forth good or repelling harm. Knowing that necessitates knowing that none is worthy of worship but Allah as well as the falsehood of any other deities.

Sixthly, the concurrence of all the books of Allah (divine scriptures) upon this fact.

Seventhly, the most special of all people, who have the most perfect characteristics, intellects, opinions, sound judgment and knowledge - and they are the Messengers, Prophets, and devout scholars - have testified this to Allah (that none has the right to be worshiped except Him).

Eighthly, the evidence that Allah placed in the universe and in the human soul and which are the greatest proofs of the Oneness of Allah, that proclaim Tawheed (pure monotheism) with what He has laid in them of His subtle work, brilliant wisdom, and His marvelous creation.

These are the ways that Allah, The Exalted, often employs to demonstrate to the creation that none has the right to be worshiped except Him, and He manifests them throughout His Book and reinstates them when the slave contemplates over them. Through them, the slave inevitably attains certainty and firm recognition about the testimony of faith (Laa ilaaha illa Allah). How much more if all of the signs of Tawheed were combined and manifested from all angles; that would without doubt deeply implant faith in the heart of the slave, so much that it will be as firm as solid mountains, unshakeable by doubts and ambiguities or by exposure to falsehood; rather, it will increase in growth and perfection. If you were to look at the mightiest evidence and the grand affair, which is done by studying this Noble Quran and contemplating its verses, you will realize that it is the greatest way to learn about the Oneness of Allah, and the slave attains by means of it (the Quran) what he does not attain by any other means.”

Allah knows best.