当你中断内部循环并开始外部循环的下一次迭代时，csv_compare不会重置到开始，它会从你停止的地方继续，一旦你用完了迭代器，就结束了。
您需要在外部循环的每次迭代的顶部重置迭代器，这最容易通过简单地打开那里的文件来完成。

with open('input.csv', newline='') as csvfile, open('output.csv', 'w' ,newline='') as OutputCSV:
    csv_input = csv.reader(csvfile)
    csv_output = csv.writer(OutputCSV)
    csv_output.writerow(next(csv_input))

    for row in csv_input:
        with  open('compare.csv', newline='') as keyCSVFile:
            csv_compare = csv.reader(keyCSVFile)
            for entry in csv_compare:
                if row[0] == entry[0]:
                    csv_output.writerow(row)
                    break

我建议从csv_compare读取第一列到list或set，然后只使用单个for循环：

import csv

finalCSV = {}
with open("input.csv", newline="") as csvfile, open(
    "compare.csv", newline=""
) as keyCSVFile, open("output.csv", "w", newline="") as OutputCSV:
    csv_input = csv.reader(csvfile)
    csv_compare = csv.reader(keyCSVFile)
    csv_output = csv.writer(OutputCSV)
    csv_output.writerow(next(csv_input))

    compare = {entry[0] for entry in csv_compare}   # <--- read csv_compare to a set

    for row in csv_input:
        if row[0] in compare:     # <--- use `in` operator
            csv_output.writerow(row)

您可以完全跳过内部循环。当第一列与compare.csv中的任何第一列值匹配时，您可以添加input.csv中的行。因此，将这些值放在一个集合中以便于查找。

import csv

with open('compare.csv', newline='') as keyCSVFile:
    key_set = {row[0] for row in csv.reader(keyCSVFile)}

with open('input.csv', newline='') as csvfile, open('output.csv', 'w' ,newline='') as OutputCSV:
    csv_input = csv.reader(csvfile)
    csv_output = csv.writer(OutputCSV)
    csv_output.writerow(next(csv_input))
    csv_output.writerows(row for row in csv_input if row[0] in key_set)

del key_set
print('wait...')