Whoever can look upon that mother's grief without springing to her relief has a heart of stone.

【翻译】旁观那位母亲的悲痛而不使她感到宽慰的人，都是铁石心肠。

spring to one's relief  跟 come to one's relief 是一样的结构和含义。就是一种动宾关系：使某人宽慰；让某人宽慰。

类似用法，如：

When one man is in difficulty, all come to his help.  

一人有难，众人相助（他）。

When I heard that he was in trouble, I came to his help at once.

当我听说他有麻烦的时候，我立刻就来帮助他了。