Acts 3:1-5
Acts 3:1 (KJB)
Now Peter and John went up together into the temple at the hour of prayer, being the ninth hour.
The hour of daily prayer in the temple was about 3 PM. Peter and John had gone to the temple for the time of prayer. In the temple was a court of the women and a court of the Gentiles. Both of these groups were only allowed to go so far. In fact, the Pharisees had used the court of the women to hold their teaching sessions so this way anyone could attend without defiling the temple.
Acts 3:2 (KJB)
And a certain man lame from his mother's womb was carried, whom they laid daily at the gate of the temple which is called Beautiful, to ask alms of them that entered into the temple;
Here begins the narrative of the lame man who was over forty years old. (Acts 4:22 KJV) For the man was above forty years old, on whom this miracle of healing was showed. He was lame from the time of birth and so he had to be carried to the gate. No doubt it was probably the family members who carried him to the gate and then brought him home every day. He was placed at the entrance to the temple where he could ask for alms or charity. In those days there was no such thing as disability insurance. A person who was lame or disabled had to beg and make their living on the generosity of others. It is not sure which gate he was laid at because there were nine gates but it is believed that it was the Eastern Gate which may have been the busiest. I am sure his family would have placed him where the most people would see him and maybe bring in the largest amount of charity possible which would go to his family to help feed, clothe, and house him.
Acts 3:3 (KJB)
Who seeing Peter and John about to go into the temple asked an alms.
Peter and John was just about to go into the temple when the lame man had asked alms of them. Alms was an act of charity which showed mercy or sympathy. This could be understood as the Good Samaritan helping out the injured man and offering to pay for further care. Now it is not sure if Peter and John were walking in alone or if they were picked out of a group but nevertheless the lame man focused on them. It is also interesting to note that since this man was over forty years old, Jesus must have seen him at the temple and allowed him to remain lame until this time which would turn out to be another testimony of the faithfulness of God.
Acts 3:4 (KJB)
And Peter, fastening his eyes upon him with John, said, Look on us.
Fastening - Looking very intently
Peter had looked very intently at the man and told him to look upon him and John. The word “look” is in the Imperative Mood making it a command. Most people would not look upon these lame beggars because in the mindset of the Jews, those who were in this condition were considered to be under the curse of God. So it would have been a strange occurrence to this man that they actually stopped and looked at him. The beggars were used to people just dropping a few coins and going their way but this is not the way of Christ. (Luke 8:46 KJV) And Jesus said, Somebody hath touched me: for I perceive that virtue is gone out of me.  In a large crowd, the Lord Jesus Christ had focused on one woman who touched Him. Jesus shows that He is compassionate on every single child of His and that large crowds do not separate them.
Acts 3:5 (KJB)
And he gave heed unto them, expecting to receive something of them.
Gave heed - Give attention to or hold fast
Expecting - Wait for or anticipate
Peter commanding the man to look at him elicited a response from him of intense looking. The man was expecting to receive something from them, since he was commanded to look, maybe he thought that along with a few shekels, he was going to be chided for begging. I am sure that those who begged with physical problems were probably the source of ridicule to some and he might have expected that.