Paul wrote this letter to Hebrews to encourage them in the day of
trials. And one of the phrase that stands out in the passages is "let
us". Paul used this phrase 12 times. Than what did Paul encouraged
Hebrews by saying "let us" repeatedly through out the letter?

This phrase first came out in chapter 4 verse 1, in the context about
the "rest". And also in chapter 4 verse 11. What did Paul taught
Hebrews about the rest? To find out the answer for that question, we
first have to know what the rest is. By reading all the passages in
this letter that include the word "rest", we can tell that Paul used
the word "rest" to explain about two thing. One is the "rest" in
Mosaic era (3:16-19). And the other is the "rest" in the New Covenant,
the rest that we are promised to enter (4:1).

"Therefore, while the promise of entering his rest still stands, let
us fear lest any of you should seem to have failed to reach it."
- Hebrews 4:1

The Israelites in Mosaic era did not obey the Lord and sinned, so God
was provoked by their sinful behavior . So the Israelites were not
allowed to enter the rest for their unfaithfulness. They abandoned the
promised rest by their evil deeds. So Paul encouraged Hebrews and us,
not to follow the bad example of Israelites but to keep the faith and
draw near to the Lord like our faithful ancestors lived their faithful
life (11:1-40).

"So we see that they were unable to enter because of unbelief."
- Hebrews 3:19

There were also people that followed the bad example of Israelites.
People who fell away from God with their unbelieving heart (3:12),
those who well by disobedience (4:11), and the people who fallen away
from the faith (6:4-6). They are the one who did not strive to enter
the rest. And they were one of the problems in the days that Paul
wrote this letter. So Paul wrote the letter to Hebrews to encouraged
them to have hope to receive what is promised and to enter the rest
(5:11, 6:18-19). One of the causes of this problem was the trials that
they were facing. They were suffering for Christ's sake.

What Paul wrote to them is to be filled with joy especially in the
trials to endure the suffering (10:34). Because this is what they did
when they were persecuted before. Not only themselves, but also the
fathers who patiently endured in the trials to receive the promise.
Paul wrote about the example of the fathers in chapter 11 very
specifically by writing about the story of 10 faithful ancestors. With
these things in mind, we can conclude that one of the things that Paul
encouraged the Hebrews with the phrase "let us" is to strive to enter
the promised rest and to keep the faith.

"Therefore, since we are surrounded by so great a cloud of witnesses,
let us also lay aside every weight, and sin which clings so closely,
and let us run with endurance the race that is set before us..."
- Hebrews 12:1

What else did Paul encourage the Hebrews in this letter? The phrase
"Let us ... draw near to the throne of grace (4:16, 10:23)" comes out
twice in this letter. What was the purpose of this phrase? By reading
the context, we can tell that "to draw near" is the commandment in the
New Covenant. Because in the Old Covenant, even the high priests were
allowed to enter the Most Holy Place once a year. And people of Israel
were not allowed to draw near to God. Because the promise of the
eternal redemption was not full filled yet by Jesus Christ. This is
the reason that even the high priests were allowed to drawn near to
God in limited occasions. But by the perfect forgiveness for our sins,
the sinners were allowed to draw near to God (4:16, 10:19). By the
blood of Jesus Christ and the sanctification, the unclean sinners were
purified so that we can drawn near to God boldly with faith.

What are we going to do by drawing near to God? In the Old Covenant,
the high priest entered the Most Holy Place to offer the offerings
(9:1-10). Also the priests and Levites were serving at the House of
the Lord for the "temporal" redemption (10:25-11:4). But after Jesus
Christ full filled the promise of eternal redemption, We are now
allowed to go into the true tabernacle to offer the sacrifice of
praise (13:15). And this is the grace that was given to us through
Jesus Christ. Paul encouraged the people so that no man will fail from
the grace of God (12:15).