First, let's look at the aya and the explanation:

Those whom Allah (in His plan) willeth to guide,- He openeth their breast to Islam; those whom He willeth to leave straying,- He maketh their breast close and constricted, as if they had to climb up to the skies: thus doth Allah (heap) the penalty on those who refuse to believe. (Quran 6:125)

Allah's Universal Plan is the Qadha wa Qadr, which is so much misunderstood. That Plan is unalterable, and that is His Will. It means that in the spiritual world, as in the physical world, there are laws of justice, mercy, grace, penalty, etc., which work as surely as anything we know. If, then, a man refuses Faith, becomes a rebel, with each step he goes further and further down, and his pace will be accelerated; he will scarcely be able to take spiritual breath, and his recovery -in spite of Allah's mercy which he has rejected-will be as difficult as if he had to climb up to the skies. On the other hand, the godly will find, with each step, the next step easier. Jesus expressed this truth paradoxically: "He that hath, to him shall be given; but he that hath not, from him shall be taken away even that which he hath": Mark, 4:25. John (6:65) make Jesus say: "No man can come unto me, except it were given unto him of my Father."

Without understanding the peculiarities of the Arabic language, it is indeed difficult to interpret this aya. In fact, it seems from the Hungarian text as if there was a predestination in Islam. However, there is no any predestination in regard of the individual's free choice. So, the fate has two sides. There is a predestined part that man cannot change. For example, what we are born into, like wealth, poverty, unexpected adversity, earthquake, volcano eruption, etc. However, it is up to me to decide which way to choose in poverty, steal or go to work? The same applies for faith. Until the word of Allah has not come to me, I'm innocent. But if it gets there and I have a choice to make a decision, there is no predestination on how to decide. I'm responsible for my decision. In the Arabic text, "Those whom Allah (in His plan) willeth to guide..." refers to the word "yahdi", guide, lead. It is understood that Allah wants to lead and divert, but this is rather a test. The term "yahdi" occurs several times in the Quran. The above aya is better understood if I give two more examples on the behavior of the word "yahdi" in the following ayas:

It is true thou wilt not be able to guide every one, whom thou lovest; but Allah guides those whom He will and He knows best those who receive guidance. Quran 28:56)

The immediate occasion for this was the death of Abu Talib, an uncle whom the holy Prophet loved dearly and who had befriended and protected him. The Prophet was naturally anxious that he should die in the profession of the true Faith, but the pagan Quraish leaders persuaded him to remain true to the faith of his fathers. This was an occasion of disappointment and grief to the Prophet. Allah guides whom He wants or Allah guides the one who wants Him. Both interpretations are possible from the text.

If Allah so willed, He could make you all one people: But He leaves straying whom He pleases, and He guides whom He pleases: but ye shall certainly be called to account for all your actions. (Quran 16:93)

Allah's Will and Plan, in allowing limited free-will to man, is, not to force man's will, but to give all guidance, and leave alone those who reject that guidance, in case they should repent and come back into Grace.
We also find as previously the active and passive form of His guidance. 1. Active: He guides whom He wants. 2. Passive: He guides the one who wants Him. According to the Arabic text both are correct. I agree with the latter one.

And here's the point. Since the word "yahdi", guide, can be both active and passive in the Quran, it refers to the duality of fate. The predestined and the free will part. Accordingly, the interpretation of aya is correct as follows:

Those whom Allah (in His plan) willeth to guide,- He openeth their breast to Islam (provided he openeth his breast by his own will); those whom He willeth to leave straying,- He maketh their breast close and constricted, as if they had to climb up to the skies (provided he closeth and constricth his breast by his own will): thus doth Allah (heap) the penalty on those who refuse to believe. (Quran 6:125)

Answer: God gives the grace of faith and makes his path easier for the one who wants it. He does not predestine this path for anyone. If anyone comes close to Islam, knows the teachings, he/she should decide freely how to keep going.