What was the British public's reaction to a German who didn't speak English coming over to rule in 1714?

Thackeray, despite being a 19th century author, uncritically passes on the general view of the time: "His heart was in Hanover... we took him because we wanted him, because he served our turn; we laughed at his uncouth German ways, and sneered at him. He took our loyalty for what it was worth; laid hands on what money he could; kept us assuredly from Popery... Cynical and selfish, as he was, he was better than a king out of St. Germains with the French king's orders in his pocket, and a swarm of Jesuits in his train."

George was widely ridiculed and criticized. His subjects criticized him for nearly everything. They criticized his two mistresses for not being pretty enough. They criticized him simply for being German. They criticized him for never learning to speak English, even though we know he read, wrote, and spoke it perfectly by the latter half of his reign. His treatment of his wife was considered a scandal (and, to be fair, was genuinely bad). He was considered wooden and unpersonable, which sometimes became accusations of stupidity.

However, despite this, George I had wide support as monarch, even from the start. While most of the English didn't like him personally, they universally acknowledged his Protestantism and respect for constitutional government. He also managed to keep things relatively stable. This was important in a country that had just undergone half a century of civil wars and coups over who got to be monarch. He was considered good for England, even if not personally popular, and he ruled a country that was secure, stable, prosperous, and free.

Were they used to foreign rulers by that point?

England had been ruled by Scottish Stuarts since 1603, with a brief interruption of William and Mary (a Dutch Prince and his Stuart wife). The Stuarts also married other royals, meaning Charles I, Charles II, and James II all had foreign mothers (James I's parents were both Scottish, and Queen Anne's parents were both English). But the dynasty was largely English in character, spending most of its time in London. Additionally, Scotland and England were culturally similar by this point, a process accelerated by the personal union. For example, the Scottish and English elite languages were somewhat mutually intelligible.

So while England was, in some sense, used to foreign royals and rulers who might not be perfectly English, they were not used to the King himself being radically non-English. There was the example of William of Oranje, but he spoke English, had close ties to England, and had married an Englishwoman. George I, in contrast, was widely seen as a completely alien person. This was much commented on at the time, as seen above.

However, George I had one great advantage: he was Protestant. In fact, he was chosen because he was the closest relative to the King who was Protestant. This was seen as an even more important qualification than being English. Parliament passed over about fifty Catholics, some of whom were English, to give him the throne. He was Lutheran, yes, and the Anglican Church was more Reformed, but such disputes were less important than the Catholic-Protestant distinction.

Did this hamper George I's ability to rule in any way?

George was not entirely German himself. He had a claim to the crown through his grandmother, Elizabeth of Bohemia, who was English. (But married the Winter King, and thus gained 'of Bohemia'.) Also, it's clear by about 1720 George could speak English and not entirely clear that he couldn't beforehand. He was also fluent in French and Latin, both of which would have been common at court. So it's unlikely he had real trouble communicating with his ministers

This was an age before mass politics and he would not have often been expected to appear in public. George I was particularly loathe to do so and usually declined to give any speeches, even to Parliament. This was notably in character with George I in general. George I hated being in public or watched, even in private non-formal settings. He didn't even like people looking at his carriage as it traveled.

George was not an ineffective monarch. However, England was already largely a constitutional monarchy and George I accelerated the process. George I and his ministers basically created the political regime that lasted in Britain for at least a century, and arguably continues to this day. His embracing of constitutional monarchy and reorganizing government to put more power in ministerial hands is sometimes seen as personal or political weakness (or apathy). Yet it's apparent from the records of the time these powerful ministers, while pursuing their own agendas, understood themselves to serve at the King's pleasure. George I took an interest in policy and influenced it. It's also apparent George was purposefully empowering these ministers rather than being forced to against his will.

It is possible George I was 'seeing the writing on the wall' and deciding to go with the flow rather than fight the current. That's another question. Certainly George I could have, like his predecessor Queen Anne, tried to take a more active role in affairs. He chose not to, and largely achieved his policy objectives without exerting such direct power.